INTRODUCTION
There is a growing interest for questions pertaining to wave spread in disordered
lattices, which are related to the search of optical or acoustic localization
and recently of cold atom localization (Economou and Alkire,
1988; (John and Stephen, 1983; Skipetrov
et al., 2008). The wellstudied case of electronic systems with independent
site disorder does not fully cover all cases of such wavelike excitations in
complex media. A wellknown result of the Anderson model for the site energy
is the absence of long range transport in one dimensional system. All electronic
states in one dimension are exponentially localized regardless of the amount
of disorder (Anderson, 1958).
Much attention has been paid to special disorder correlations for which new
phenomena are expected to appear. For instance, although Anderson localization
occurs in one dimension, one finds partial delocalization even for an infinitesimal
amount of disorder in the presence of correlations (Datta
et al., 1993; Sanchez et al., 1994).
A number of recent works dealing with tightbinding Hamiltonian strongly suggest
that the occurrence of correlations in neighbour random parameters are not independent
with a correlation length (Evengelou, 1990), (Dunlap
et al., 1990; Wu and Phillips, 1991). Furthermore,
the existence of a mobility edge between extended and localized states was found
for 1D random system with weak longrange correlated disorder) (Molina,
2005; Esmailpour et al., 2006). Longrange
disorder induces the appearance of delocalization and long range transport.
The Random Dimer Model (RDM) can be shown to be an example of the correlated
disordered system. In this 1D random model, the site energy takes one out of
two possible values, one of which is distributed at random to pairs along the
chain, so that the correlation length coincides with lattice spacing. On the
basis of this interest the authors claimed that the RDM has states
which are extended over the whole sample, with N the number of sites in the
system. A discrete number of extended states was found numerically (Evengelou
and Wang, 1993), (Evengelou and Economou, 1993)
and was observed recently in the experiment with semiconductor random superlattices
(Bellani et al., 1999). In Kronig Penney model,
the electronic field delocalizes the eigenstates where the wave functions decay
with a power (Soukoulis et al., 1983; Cota
et al., 1985), in this regime, the resistance was checked experimentally.
For sufficiently large field strengths, the eigenstates become extended (Markos
and Kramer, 1993; Markos and Henneke, 1994). When
the electric field vanishes, it is wellknown that the spectrum is then pure
and dense), (Abrahams et al., 1979; Landauer,
1970). The wave functions are exponentially localized, with a localization
length that decreases with increasing disorder (Mott, 1968).
The transmission coefficient has been used successfully to analyze the nature
of the electronic states. The effect of exponentially localized eigenstates
can be observed in the exponential decreases of the transmission coefficient
with the length of the system (Anderson et al., 1980;
Thouless, 1974). Moreover, the connection between resistance,
or more precisely, conductance and transmission coefficient can be carried out
via Landauer formula (Landauer, 1970). One find:
with ρ the resistance. Such a behaviour can be expected from the selfaveraging of the Lyapunov exponent γ which is the inverse of the localisation length in 1D systems. This characteristic length l is defined as:
This parameter is always positive and describes the spatial scaling properties
of a disordered system (Soukoulis and Economou, 1981;
Liffshitz et al., 1988).
In this study, we first discussed the delocalization induced by correlations in the Kronig Penney model. Here, we used an array of δfunction potentials with independent random strengths and study numerically the transmission properties for a finite length of the lattice. We derived exact results for the main characteristics of the model using a transfer matrix combined with a Poincaré map approach. Secondly, we examined the size dependence of the transmission coefficient of a linear RDM chain subject to electric field. When F = 0, the transmission coefficients at particular energy close to 1 and a deep minimum around the resonant energy in the resistance is found, indicating that the localization length of those states is large. For F ≠ 0, we observed that the transmission decreases with increasing F where F is the electric strength. However, this minimum disappears and the values of resistance become extremely large. As soon as F is present, the electron gains the energy V(x): V(x) = Fx. This electrical potential suppresses the resonance energy induced by correlation. That induces a transition between extended and localized behaviour. Finally, we discussed our calculations of the Lyapunov exponent that indicate that all states around the resonance in the presence of the electric field have a localization length smaller than the system size. We expected from this result a mobility edge that depends on the strength of the field in the RDM.
MODEL
Here we considered a 1D KronigPenney model with random δfunction potentials subject to an applied electric field. The problem is defined by the Schrφdinger:
where, λ_{n} is a set of independent random variables that measures the strength of the δpotentials. Here E is the energy of the electron measured in atomic units and Ψ is the wave function. We proceed with the problem of the disordered lattice containing a certain number of pair impurities placed randomly. We kept the positions of the δfunctions to be regularly spaced {x_{n} = n} but we introduced a correlated disorder, for which λ_{n} takes only two values: λ and λ', where appears only in pairs of neighbouring sites (dimer impurities). The electronic potential V(x) is given by Fx term in Eq. 1 with Fdenoting the electric field strength..
In this section, we presented a numerical study of the transmission coefficient
of this model. Our approach is inspired by Soukoulis et
al. (1983), Flores et al. (1989) who
investigated the transmission coefficient and the nature of the electronic states
in 1D disordered systems. They found that the transmission coefficient behaves
as, with .
This reveals powerlaw localization.
Here, we calculated the transmission coefficient Tin the above model (RDM)
using the transfer matrix approach. We took an electron impinging from the left
of a set of δfunction potentials with wave function Ψ_{0}(x)
= e^{iqox}+rne^{iqox}. The energy of the electron is E = q^{2}_{0}
with q_{0} the momentum of the incident electron. The wave function
in the righthand side of the sample of length L is Ψ(r) = T_{N}
e^{iqrx} Here
with L = N+2; q_{r}denotes the momentum of the emerging wave. t_{N
}and r_{n} are the transmission and the reflection amplitudes of
the RDM with N scatterers respectively between two impurities, we will replace
V(x) by a constant value so that the solution between two impurities are plane
wave functions (Soukoulis et al., 1983; Cota
et al., 1985).
However, this is valid only when the electric potential between the ends of a sample is infinitesimally small.
The solution of Eq. 1 can be computed recursively for both
transmission and reflection amplitude using wellknown transfermatrix technics
(Kirilov and Trott, 1994). Then, the transmission amplitude
can be written as:
where A_{n} ≡ t_{n} and:
Equation 2 supplied two boundary conditions, A_{0} = 1 and A_{1} = 1 to determine the amplitudes completely. q_{n} is the momentum of the electron at the site n. Finally, the transmission coefficient can be calculated for each chain from:
RESULTS AND DISCUSSION
We first discussed our numerical results on the transmission coefficient for a RDM and investigated what changes occur when the electric field is applied along the lineair chain.
We choose for convenience the length L = 1000 and a dimer concentration equal to 20\%. We fixed λ =1 for the values of potential strength of the host lattice and λ′ = 1.5 for the dimer impurities.
Present results are similar to the ones obtained in by Sanchez
et al. (1994), Dunlap et al. (1990),
where a unique energy was found in the allowed band (recall their model is a
single band) and where a perfect transmission T = 1 was see in the RDM. In such
case, the system of electronic transport becomes ballistic. Thus, nondecreasing
transmission coefficient for particular energy shows the existence of extended
states arrown this one.
In Fig. 1, we showed the transmission coefficient versus energy for intervals near the first resonance. The spectrum of the Kronig Penney model follows the equation 2q cos q+λ sin q≤1 (this is the condition to be able to move in the perfect lattice) when λ is fixed. Here, we have averaged the transmission coefficient for 1000 realizations with an accuracy of 1\%. We found that around the first resonance E_{r} = 3.75 the transmission coefficient reaches values very close to 1. All realizations show the same peak around E_{r}. It is clear from Fig. 1 that the states close to the resonant energy have good transmission properties, similar to those of the resonant energy.
When F ≠ 0 , there are some important differences with respect to the case F = 0. For the same concentration of dimer impurities, we observe that the transmission coefficient decreases for a field as small as 5.10^{4}. In this case, the small F will only slightly shift the resonant energy and the transmission coefficient will completely vanish.
We showed in Fig. 2, the resistance of a RDM in both the
presence and the absence of an electric field with the same concentration of
impurities. The lower curve, corresponding to a dimer model with F = 0, exhibits
a minimum resistance about ten orders of magnitude below the resistance for
10^{4} (the middle curve). For the F considered in Fig.
2, the curve saturates to essentially a constant value with energy. However,
the resistance becomes extremely big compared to the same RDM without electric
field.

Fig. 1: 
Plot of resistance versus energy in RDM, with L = 10^{4}
for different values of the electric field 

Fig. 2: 
Plot of resistance versus energy in RDM, with L = 10^{4}
for different values of the electric field 
Such as the localization scenario is quite different in the presence of an
electric field.
The dependence of the resistance with system size is useful to study the spatial structure of the electronic states. Exponentially localized states lead to a nonohmic behaviour of the resistance, which increases exponentially with the system size.
In Fig. 3, around the resonance, the resistance has a constant value which indicates that the band of state exist with very good transport for a dimer model without field. In this case, the effect of the correlation in the random dimer potential is dominant, the electron gains more kinetic energy behaving essentially as a free particle in a potential well.
Not only the resonant energy has a low resistance for any length of the chain (lower curve), but also when F is far from zero F = 10, the plot shows a relatively small resistance, exhibits a good behaviour (middle curve). For F = 5.10^{4} and for large L the resistance rises quickly to large values.
To investigate the nature of electronic states around the resonance, we have analyzed the average scaling of In T with the system size.
In Fig. 4, we showed the results for ⟨In T⟩ versus
L for a fixed value of the energy and for different values of field. First,
as was done for the resistance, we compared our results to the size dependence
of transmission coefficient when the electric field is present. We saw that
for F = 0 the curve is flat and ⟨In T⟩ reaches a constant value. We
concluded that the states are extended. These extended states are not of the
Blocktype encountered in periodic solids (Hilke and Flores,
1997; Xiuqing and Xintian, 1997).
When F = 0 on the other hand, we observed three things. For small L<2000 we obtained similar behaviour as for F = 0 However, for increasing F, the value of <In T>changes considerably for relatively small changes of F which suggests exponential decreasing for transmission coefficient, with an exponent that depends on F. For L>700, the electronic states are stretched exponentiallocalized. This means that this phase has a zero measure in the thermodynamic limit. For F = 5.10^{4} this phase, will diverge for large L. Here, the system will be return to equilibrium.
We investigated the Lyapunov coefficient which represents the inverse of the localization length l_{c}. As is shown in Fig. 5, when F = 0, energies close to the resonant energy E_{r} have γ—10^{4}. This is in agreement with the notion that delocalization of the electronic states occurs l_{c}™10^{4}.
When we increased F, we observed an increase of the Lyapunov exponent that
stays much smaller than one. This effect coincides with the standard definition
of γ) (Liffshitz et al., 1988). The localization
can be explained by the fact that when the electric field strength is increased,
the effective potential component as FN>E_{r} indicating that the
states decay as exponentiallaw.

Fig. 3: 
Plot of the resistance ρ versus length L at the resonant
energy E_{r }For F = 0 (dark) a band of state exist with very good
transport. For F = 0, a good transport exist for L™700 (red). If F
= 5.10^{4} (blue) the resistance converges to a large value 

Fig. 4: 
Plot of < In T > versus length L at the resonant energy
E_{r} for different values of F F = 0 (dark), F= 10^{4} (red) and
F = 5.10^{4} blue) 

Fig. 5: 
Plot of Lyapunov exponent versus energy with L = 10^{4}
and for different values of strength of field F = 0 (dark), F = 10^{4}
(red) and F = 5.10^{4} (blue) 
We concluded that the delocalizationlocalization can be observed in dimer
systems such as the electric field suppresses progressively the effect of the
correlation.
CONCLUSION
We have studied the effect of electric field on a linear chain with correlated
disorder. To analyze the properties of electronic transport, we have used the
KronigPenny model. Based on the results, we have noted that the electric field
impedes the movement of the electrons in the presence of correlation. For relatively
small field, we notice that the transmission is stretched exponentiallaw decaying
with the length. This decaying depends on the strength of the electric field.
The electric field has an effect on the resonance energy that carries with it a variation of transmission coefficient which influences the nature of the electronic states. The Lyapunov exponent was also used to analyze the localization length, we have found out that when the electronic field increases, the Lyapunov exponent is saturated by a constant value that is lower then the system’s size, which indicates a localization of the electronic states.