
Research Article


Study of Defects Effect on Electronic Conductance Through Binomially Tailored Quantum Wire


H.S. Ashour
and
A.I. Assad



ABSTRACT

This study considers the effect of the defects on the
electronic conductance properties in Binomially Tailored Quantum Wires
(BTQW), in which each Dirac delta function`s potential strength have been
weight on the binomial distribution law. A single freeelectron channel
is incident on the structure and the scattering of electrons is solely
from the geometric nature of the problem. We found that this novel structure
has a good defect tolerance within Â±5% or more for the following
defects: single strength defect, dislocation defect, or both defects.
Finally, we found this structure has somehow good tolerance for flipped
order of the delta potential and missing Dirac delta potential from the
binomial pattern.







INTRODUCTION
Electronic conductance behavior in one dimensional periodic structure,
like a finite series of Dirac delta function potential, is an important
subject in condensed matter after the major advances in nanotechnology
and microfabrications. Quantum wires are one dimensional mesoscopic device,
in which the electrons can transport coherently across the whole system
with negligible inelastic scattering (Sprung et al., 1993; Singha
Deo and Jayannavar, 1994). The recent progress in modern crystal growth
techniques such as Molecular Beam Epitaxy (MBE) and the MetalOrganic
Chemical Vapor Deposition (MOCVD), has demonstrated that we can grow semiconductor
substrate with monolayer precision. These advances make it possible to
confine electrons within a lateral extent of 100 nm or less resulting
one dimensional quantum waveguide. In this wave guide, the electron transport
can be considered ballistic or quasiballistic and the electronelectron
scattering and the electronphonon interaction can be neglected if the
temperature is low enough. So, the phasecoherence length become large
enough compared with the device dimension. Therefore, the electron transport
properties solely depend on the geometrical structure of the problem in
hand. Recently, the electronic conductances in a series of Dirac delta
function potential grasp many researches interest (Sprung et al.,
2008; Ashour et al., 2006; Martorell et al., 2004; Fayad
et al., 2001; Jin et al., 1999; BoltonHeaton et al.,
1999; Ferry and Goodnick, 1999). The researcher used different methods
to study the electronic conductance through quantum wires and rings (Midgley
and Wanc, 2000; Tachibana and Totsuji, 1996; Macucci et al., 1995;
Sprung et al., 1993; Takagaki and Ferry, 1992a, b).
Recently, Ashour et al. (2006) has proposed a novel structure
which is the Binomially Tailored Waveguide Quantum Wires (BTQW), in which
each Dirac`s Delta function potential strength has been weighted on the
binomial distribution law. In this study, we study the defects effect
on the electronic conductance on the novel structure proposed by Ashour
et al. (2006).
TRANSMISSION THROUGH PERIODIC STRUCTURE
Here, let us consider a finite periodic structure of Dirac delta
function potential (Dirac Comb). Also, we have assumed that the structure
is narrow enough so that just single channel of electrons can be considered.
In this treatment, we assume the temperature is low enough to ignore the
electronelectron interaction and electronphonon interaction. We assumed
the scattering of electrons mainly form the geometrical structure of the
potential. The potential can be written as:
where, U_{j} and x_{j} represent the strength and
the position of the jth delta function respectively and N is the number
of the Dirac delta functions in Dirac Comb. The distance between the adjacent
barriers are given by d_{j} = x_{j+1}–x_{j}.
The Schrödinger wave equation in one dimension can be written as:
where, V(x) is the periodic potential given by Eq. 1,
m* is the electron effective mass, which is considered approximately constant
over the interaction range. The solution of Schrodinger wave equation
for single Delta function potential can be found in literature and also
the transfer matrix formulism (Kostyrko, 2000; Sheng and Xia, 1996; Wu
and Sprung, 1994; Wu et al., 1991; Merzbacher, 1997). The transfer
matrix for periodic structure has been used also to study the transmission
of electron through Comb structure (Ashour et al., 2006; Fayad
et al., 2001; Kostyrko, 2000; Sheng and Xia, 1996; Wu et al.,
1991). In the following, we are going to outline the matrix transfer matrix
method and we are closely following the references (Landau and Lifshitz,
1981). To derive the transfer matrix for the jth Delta function potential,
we express the electron wave function in the leads, where the potential
is zero, as:
for the left lead (x_{j–1}<x<x_{j}) and
for the right lead (x_{j}<x<x_{j+1}).
Thus and
the wave amplitudes on either side of the jth Dirac delta function after
imposing the boundary conditions satisfies,
In this expression T_{j} is the transfer matrix,
Thus β_{j} is γ_{j}/2k, where γ_{j}
is 2m*U_{j}/η^{2
}

Fig. 1: 
Conductance spectrum G in the units of 2e^{2}/h
as a function of kd/π for a sequence of Dirac delta function
potential with N = 10. The strength of the potential here is Ω
= 0.2. Notice that the number of ripples in the allowed band is N–1 
The transfer matrix for Dirac comb, which is a series of equally spaced
Dirac delta function potentials, has the following form:
and is given by the product of the transfer matrices of each individual
Dirac delta function potential
Then the transmission amplitude is given by Sprung et al. (1993):
thus is
the second element in the second row in a 2x2 matrix. The electron conductance
through this structure, according to the LandauerButtiker formula, is
(Landau and Lifshitz, 1981; Baym, 1974):
We rescale the strength of the Dirac delta function potential by the
following parameter (Takagaki and Ferry, 1992b) .
In Fig. 1, we show the conductance through N = 10 Dirac
delta function potential with strength Ω = 0.2. A perfect transmission,
in this case, is in general impossible as predicted by Ashour et al.
(2006) and Blundell (1993). According to Blundell (1993) we can not have
a resonant transmission, T = 1, even if N is very large.
BINOMIALLY TAILORED DIRAC DELTA FUNCTION POTENTIAL
Here, we reintroduce a novel simple structure of the waveguide quantum
wires based on the binomial distribution (Ashour et al., 2006),
this propose a new structure based upon the similarities between the electromagnetic
waves and the electronic plane waves. We have noticed a similarity between
the diffraction of plane waves from multiple narrow slits and the electrons
diffraction from Dirac comb. Because of this situation, we introduced
a novel simple structure of the quantum waveguide based upon the binomial
distribution (Fig. 2) to get tunneling transmission
to reach the unity over a significant range of incident electron energy
and to get rid of the undesired ripple in the conductance band in the
previous structure. The Dirac delta function has been equally spaced but
their strength, Ω_{j} and has been weighted according to
the binomial distribution law, which is:
Thus, Ω(N_{j}) represents the strength of the Dirac delta
potential. N+1 represents the total number of Dirac delta function potentials
in the quantum wire and N_{j} represents the order of the Dirac
delta potential. This novel structure of quantum wires can be realized
by putting metallic gates on top of a onedimensional electron gas and
then by applying voltages, according to the binomial distribution law,
to deplete the electron gas underneath. In this case, Eq. 8 is no longer
valid for our new structure so that, the total transmission matrix can
be written as follows:
Notice that the potential strength is weighted according to Eq.
11. In Fig. 3, we show the conductance
spectrum through a sequence of binomially tailored Dirac delta function
potentials. It is

Fig. 2: 
Binomially tailored Dirac delta function potential.
Here, N = 4 but the number of Dirac delta functions is 5. and N_{j}
values which can be evaluated by Eq. 11 

Fig. 3: 
Conductance spectrum G in the units of 2e^{2}/h
as a function kd/π of for a binomially tailored sequence of Dirac
delta function potential with N = 4. The strength of the potential
here is as in Fig. 2 
quite interesting to notice that the transmission through this structure
approaches unity in the allowed band region without any ripples after
some small values of k. We have a resonant tunneling due to coherent interference
due to elastic scattering of electrons plane waves, which leads the transmission
to reach unity over a considerable range of k values, which is called
allowed band or conduction band. Also, we can see that there is a forbidden
band, or a conduction gap where the transmission is small.
In Fig. 3, we show the conductance spectrum through a sequence of a binomially
tailored Dirac delta function potentials. It is quite interesting to notice
that we have reached a transmission through this structure approaches
to unity in the allowed band region without any spikes after some k value.
Here, we have a resonant tunneling due to coherent interference effects
due to elastic scattering of electrons, which leads the transmission to
reach unity and also to have constant value over the allowed band or conduction
band. Also, we see that there are forbidden bands or conduction gap where
the transmission is small.
DEFECT EFFECT ON THE ELECTRONIC CONDUCTANCE
Strength Defect
In this subsection, we study strength defect effect on the central element
of the binomial tailored quantum wire and keeping the other elements and
the spacing between the Dirac delta function potentials intact. First,
in Fig. 4a, we consider defect free binomially tailored
quantum wire with N = 35, with two scaling factors. We notice that when
the scaling factor increase the conduction bands become narrower but the
forbidden bands become wider and well defined. In Fig.
4b, we consider the strength defect does not exceed ±5% of
the Dirac delta function potential strength. That is, when the central
Dirac delta function potential strength is, for odd number of Dirac delta
function potential in the binomial distribution. In Fig.
4b, we plot the electronic conductance spectrum for both strengths
(with scaling factor of one and three) with N_{j} is 35 and scaling
factor of three. As can noticed there is slight difference between the
two curves, and compared to Fig. 4a. In Fig.
4c, we increase the strength defect up to ±20%, we have noticed
some measurable differences between the two curves and compared to Fig.
4a, we have noticed some measurable differences between the two curves,
but still the conduction band and the forbidden bands are well defined,
which is a very good feature for the binomially tailored quantum wires.

Fig. 4a: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. The curve with squares is for N = 35 without
scaling the binomial distribution. The second curve also with N =
35 but with a scaling factor of three 

Fig. 4b: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. Here, N =35 with scaling the binomial
distribution by factor of three. In the case where the defect is only
±5%, in the strength of the central Dirac delta function, there
is slight difference between the two curves and compared to Fig. 4a 

Fig. 4c: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. Here, N = 35 with scaling the binomial
distribution by factor of three. In the case where the defect is only
±20%, in the strength of the central Dirac delta function,
there some measurable differences between the two curves and compared
to Fig. 4a. The curve with circles, has potential strength defect
+20% higher 
Dislocation Effect
In this research, we study dislocation defect effect on the position
of the central element in the binomial tailored quantum wire and keeping
all other elements and the spacing between the Dirac delta function potentials
unchanged. First, We consider the position defect does not exceed ±5%
of the Dirac delta function potential spacing constant. That is, when
the central Dirac delta function potentials spacing is d = ±0.05
d. In Fig. 5a, we plot the electronic conductance spectrum
for both dislocations with N = 35 and scaling factor of three. Compared
to Fig. 4a, as can noticed there is a difference between
the two curves. The conduction band starts to lose its flatness and the
forbidden band become sharper spacing increase between the central Dirac
delta function and the adjacent one. In Fig. 5b, we
increase the dislocation defect up to ±20%, we have noticed measurable
differences between the two curves and the curve in Fig.
4a, but the conduction band is still well defined but the forbidden
bands have a split compared to forbidden band in no defect curves (Fig.
5). This splitting is due to resonant state in the forbidden energy
band which leads to a bound state in the structure (Singha Deo and Jayannavar,
1994). This is because the particle mode cannot propagate and hence get
trapped.

Fig. 5a: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. Here, N = 35 with scaling the binomial
distribution by factor of three. In the case where the defect is only
±5%, in the position of the central Dirac delta function, there
is some difference between the two curves and compared to the curve
in figure 4a. The curve with circles, dislocation in position +5%
wider 

Fig. 5b: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. Here, N = 35 with scaling the binomial
distribution by factor of three. In the case where the defect is only
±20%, in the position of the central Dirac delta function,
there some measurable differences between the two curves. The curve
with circles, dislocation in position +20% wider 
Dislocation and Strengt
In this research, we study both the dislocation defect and the strength
effect on the central element in of the binomially tailored quantum wire
and keeping the other elements strength and spacing in between intact.
In Fig. 6a, we plot the electronic conductance spectrum
considering the position defect is d±0.05 d and the strength defect
is Ω_{j} (N/2+1)±0.05 Ω_{j}(N/2+1). As
can be seen from the Fig. 6a there measurable difference
between the defect free structure and the structure with both defects,
but the conductance spectrum from the defected structure still maintain
the main features of the original transmission spectrum. This leads us
to conclude that this structure has a significant tolerance for both defects
in position and strength which makes this structure more reliable.
In Fig. 6b, we plot the electronic conductance spectrum considering the
position defect is d±0.2d and the strength defect is Ω_{j}
(N/2+1)±0.2 Ω_{j}(N/2+1). Here, there is a significant
and measurable difference in the conductance spectrum between the defect
free case and this case. So, we can say that this structure cannot tolerate
this high defect in both the position and strength. As can be noticed
from

Fig. 6a: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. Here, N = 35 with scaling the binomial
distribution by factor of three. In this case, where the defect is
only ±5%, in the position and the strength of the central Dirac
delta function, there is some difference between the two curves and
compared to the upper curve. The curve with triangles, dislocation
in position +5% wider and +5% higher in strength 

Fig. 6b: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. Here, N = 35 with scaling the binomial
distribution by factor of three. In this case, where the defect is
only ±20%, in the position and the strength of the central
Dirac delta function, there is some difference between the two curves
and compared to the upper curve. The curve with triangles, dislocation
in position +20% wider and +20% higher in strength. The peaks in the
forbidden bands are due to bound states 
the Fig. 6b, the conduction band conductance get lowed by this double
defect and the forbidden band has splitting. Also, this splitting is due
to resonant state in the forbidden energy band. However, the peaks increase
in its height as we increase kd/π. Increasing the defect in both
the position and strength increases the chance the particle mode not to
propagate through the structure, which increases the chance of the entrapment
as the factor kd/π increases (Wu et al., 1991).
In Fig. 6c, We plot the electronic conductance spectrum considering the
position defect is d±0.05d and the strength defect is Ω_{j}
(N/2+1)±0.2 Ω_{j}(N/2+1). In this plot, we notice
that the structure can tolerate a double defect with +5% in position and
a +20% in strength without losing the conductance spectrum pattern. From
this we can set the maximum limit of the defect tolerance of this structure

Fig. 6c: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π. Here, N_{j} with scaling the binomial
distribution by factor of three. In this case, where the defect is
only +5% in position and +20% in strength of the central Dirac delta
function. Notice that there is some difference between upper curve,
defect free Fig. 4a and curve with circles 
which is ±5% in position and ±20% in strength of the central
Dirac delta function potential in the binomially tailored quantum wire.
Missing and Reversed Order Dirac Delta Function Defect
Finally, in this subsection, we are going to study the effect of missing
and reversed order, when two adjacent Dirac delta function strength switched,
Dirac delta function potential form the binomially tailored quantum wire.
In this case, the binomially tailored quantum wire would not function
as a good quantum waveguide as it should be, as in Fig.
4a. We assumed the central Dirac delta function potential is missing
from the pattern and keeping all other elements strength and spacing intact.
Also, we have assumed the central Dirac delta function switched its strength
with the adjacent delta function. In Fig. 7a, we illustrate
and compare the effect of these defects with defectfree binomially tailored
quantum wire. In Fig. 7a, the number of Dirac delta
functions in the quantum wire pattern is seven and their strength is weighted
according to Eq.11.
As we can see, the electronic conductance through this structure is largely
affect when the central Dirac delta function is missing from the pattern
at low values of kd/π, but at high values of kd/π the conductance
spectrum starts to match that of nodefect spectrum. In the case of the
switched order, we see this structure has a good tolerance and conductance
spectrum is almost equals that is of nodefect conductance spectrum at
moderate values of kd/π. In Fig. 7b, we study
the effect of the number Dirac delta functions in the binomially tailored
quantum wire on the electronic conductance in the defected quantum wire.
We have increase the number of Dirac delta functions to eleven and their
strength weight by scaling factor of three. Its worthwhile to notice that
when the number of Dirac delta function is increased in the quantum waveguide
the difference between the no defect electronic conductance spectrum and
the one switched delta function potential case in negligible and in the
case of missing Dirac delta function potential the electronic conductance
spectrum starts to come close to the nodefect spectrum at lower values
of kd/π. As we can notice, these defects destruct of the phase coherence
of the electrons wave functions interacting with the binomially tailored
quantum wire, which leads to the irregularities in the tunneling and consequently
the electronic conductance spectrum.

Fig. 7a: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π, the number of Dirac delta function in
the quantum wire pattern is seven and the scaling factor is one. The
curve with circles is the electronic conductance through the quantum
wire without defect, the curve with squares the quantum wire has the
central Dirac delta function missing and the curve with triangles
the central Dirac delta function switched with the adjacent one 

Fig. 7b: 
The electronic conductance, in the units of 2e^{2}/h
as a function of kd/π, the number of Dirac delta function in
the quantum wire pattern is eleven and the scaling factor is three.
The curve with circles is the electronic conductance through the quantum
wire without defect, the curve with triangles the quantum wire has
the central Dirac delta function missing and the curve with squares
the central Dirac delta function switched with the adjacent one 
CONCLUSION
We reintroduced the novel structure of the waveguide quantum wires
which is the BTQW. We show that it is possible to have perfect transmission,
coherent tunneling, due the interference effects, which gives rise to
allowed band and forbidden bands in the transmission spectrum. We found
that the increase of Dirac delta function in the structure and their strength
the conduction band become wider and the forbidden bands become sharper
and narrower. Besides that, we found the structure tolerate the following
defects: up to ±20% in strength defect and ±5% in position
defect for the central Dirac delta function in the binomial distribution;
can tolerate both defect up to ±20% in strength and ±5%
in position dislocation; has a little tolerance when we replace the central
Dirac delta function potential with the adjacent one. So, we can conclude
that this novel structure offer a good electronic conductance spectrum
with considerably high tolerance for defects.

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