The paper considers the effect of the defects on the electronic transmission
properties in binomially tailored waveguide quantum wires, in which each Dirac
delta function potential strength have been weight on the binomial distribution
law. We have assumed that a single free-electron channel is incident on the
structure and the scattering of electrons is solely from the geometric nature
of the problem. We have used the transfer matrix method to study the electron
transmission. We found this novel structure has a good defect tolerance. We
found the structure tolerate up to in strength defect and in position defect
for the central Dirac delta function in the binomial distribution. Also, we
found this structure can tolerate both defect up to in strength and in position
dislocationComment: Submitted on behalf of TIMA Editions
  (http://irevues.inist.fr/tima-editions

A.I. Assad

Baym

Ferry

H.S. Ashour

Merzbacher

Midgley

English

arXiv

Crossref

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

58-62Quantum wire, electronic transmission, conductanceThe paper considers the effect of the defects on the electronic transmission properties in binomially tailored waveguide quantum wires, in which each Dirac delta function potential strength have been weight on the binomial distribution law. We have assumed that a single free-electron channel is incident on the structure and the scattering of electrons is solely from the geometric nature of the problem. We have used the transfer matrix method to study the electron transmission. We found this novel structure has a good defect tolerance. We found the structure tolerate up to in strength defect and in position defect for the central Dirac delta function in the binomial distribution. Also, we found this structure can tolerate both defect up to in strength and in position dislocatio

Shabat, M.

Ass’ad, A.-I.

Ashour, H.-S.

I-Revues

The deffect effect on electronic conductance in binomially tailored quantum wire

The paper considers the effect of the defects on the electronic transmission properties in binomially tailored waveguide quantum wires, in which each Dirac delta function potential strength have been weight on the binomial distribution law. We have assumed that a single free-electron channel is incident on the structure and the scattering of electrons is solely from the geometric nature of the problem. We have used the transfer matrix method to study the electron transmission. We found this novel structure has a good defect tolerance. We found the structure tolerate up to in strength defect and in position defect for the central Dirac delta function in the binomial distribution. Also, we found this structure can tolerate both defect up to in strength and in position dislocatio

Ashour, HS

A.I., Ass''ad

Shabat, Mohammed M.

Institutional Repository of the Islamic University of Gaza

 
 
THE DEFECT EFFECT ON THE ELECTRONIC CONDUCTANCE IN  
BINOMIALLY TAILORED QUANTUM WIRE 
 
H. S. Ashoura, A. I. Ass’adb, M. M. Shabatc 
  
aDepartment of Physics, Al-Azhar University of Gaza, Gaza Strip, P.A. 
E-mail: hashour@alazhar-gaza.edu 
bDepartment of Physics, Al-Aqsa University, Gaza Strip, P.A. 
E-mail: assad_assad@mail.alaqsa.edu 
cDepartment of Physics, Islamic University of Gaza, Gaza Strip, P.A. 
E-mail: shabat@mail.iugaza.edu.ps 
 
ABSTRACT 
 
The paper considers the effect of the defects on the 
electronic transmission properties in binomially tailored 
waveguide quantum wires, in which each Dirac delta 
function potential strength have been weight on the 
binomial distribution law. We have assumed that a single 
free-electron channel is incident on the structure and the 
scattering of electrons is solely from the geometric nature 
of the problem. We have used the transfer matrix method 
to study the electron transmission. We found this novel 
structure has a good defect tolerance. We found the 
structure tolerate up to 20%±  in strength defect and 5%±  
in position defect for the central Dirac delta function in 
the binomial distribution. Also, we found this structure 
can tolerate both defect up to 20%±  in strength and 5%±  
in position dislocation. 
 
1. INTRODUCTION 
 
In the last decade, there was a growing interest in the 
electron conductance through one-dimensional scattering 
problems, especially in those cases where the potential is 
periodic structure with finite number of identical cells [1, 
2]. Because of the remarkable advances in nano-
technology and micro fabrication, it is possible to confine 
electrons in a conductor with a lateral extent of 100nmor 
less, resulting narrow quantum wire [3]. In these 
mesoscopic devices, the electron transport is best 
described by quantum mechanics.  Miniature size of these 
devices eliminates the defect of scattering. At a low 
enough temperature, the motion of electrons through 
these devices is ballistic or quasiballistic and the electron-
phonon interaction can be neglected. So that, the phase 
coherence length enlarges enough when compared with 
the device dimension. Mesoscopic devices can be 
considered as a coherent elastic scatterer [3]. Therefore, 
the electron transport properties, solely depends upon the 
geometrical structure of the quantum waveguide.  
In recent years, there has been a growing interest in the 
electron transport through a sequence of Dirac delta 
function potential [4-9]. The researcher used different 
methods to study the electron transport in a waveguide 
quantum wire [10-12]. Recently, Ashour et al [13] has 
proposed a novel structure which is the binomially 
tailored waveguide quantum wires, in which each Dirac 
Delta function potential strength has been weight on the 
binomial distribution law. In this paper, we study the 
defects effect on the electronic conductance on the novel 
structure proposed by [13].  
 
In section 2, we outline the transfer matrix method which 
connects the solutions at the ends of the waveguide 
quantum wires. We introduce the novel structure of the 
binomially tailored waveguide quantum wires. In section 
3, we explore defect effects on the electronic conductance 
spectrum through the binomially tailored quantum wire. 
In this section, we have studied strength defect and 
dislocation defects on the central Dirac delta function in 
the distribution. Section 4 has been devoted to the 
conclusions of this study.  
   
2. TRANSMISSION MATRIX  THOUGH  
2.1  Periodic Structure 
 
In this context, we consider a finite periodic structure of 
Dirac delta function potential (Dirac Comb). Also, we 
assumed that the structure is narrow enough so that just 
single channel of electrons can be considered. In this 
treatment, we neglected electron-electron interaction, and 
we assume the temperature low enough so that electron-
phonon interaction can be neglected as well. We assumed 
the scattering of electrons mainly form the geometrical 
structure of the potential. The potential can be written as 
follows: 
( ) ( )
1
N
j j
j
V x U x xδ
=
= −∑                           (1) 
Thus, jU  and jx  represent the strength and the position 
of the j th 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 
1j j jd x x+= − . The Schrödinger wave equation of one 
dimension can be written as follows: 
 
( ) ( ) ( ) ( )
22
* 22
d x
V x x E x
m dx
ψ
ψ ψ− + =h                (2) 
  
Thus, ( )V x is the periodic potential given by equation 
(1), *m is the electron effective mass, which is 
considered approximately constant over the interaction 
range. The solution of Schrödinger wave equation for 
single Delta function potential can be found in the 
literature and also in the transfer matrix formulism [14-
16, 17]. The transfer matrix for periodic structure has 
been used also to study the transmission of electron 
through Comb structure [4-6, 14-16, 17]. The transfer 
matrix, which is related to the input electron wave and the 
output, is given by [4-6] 
 
1
10 0
10 0
j j
j j
ikx ikx
j j
ikx ikx
j j
i iC e e A
i iD Be e
β β
β β
−
− −
⎛ ⎞ ⎛ ⎞− −⎛ ⎞⎛ ⎞ ⎛ ⎞
=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠ ⎝ ⎠
 (3)  
 
Thus, jβ  is / 2j kγ , where jγ  is
* 22 /jm U h , and k  
is the wave number given by * 22 /m E h .  So that the 
transfer matrix at a given single barrier can be written as, 
 
( ) ( ) ( )1j j j jM S ikx S ikxβ−= ⋅Γ ⋅    (4) 
 
The total transfer matrix which represents the electron 
propagating through the entire device is just the repetitive 
product of the transfer matrix of a single barrier. We find    
( ) ( ) ( ) ( ) ( )
1
1
1
0t N N j j
j N
M S ikd S ikd Sβ β−
= −
⎡ ⎤
= ⋅Γ ⋅ ⋅Γ ⋅⎢ ⎥
⎣ ⎦
∏  (5) 
Thus, jd  is the periodic spacing between two adjacent 
Dirac Delta functions.  
Then the transmission amplitude is given by [18], 
( )
1
2,2t
T
M
=                   (6) 
Thus, ( )2,2tM is the second element in the second row 
in a 2 2× matrix.  According to the Landauer-Buttiker 
formula, the electron conductance through this structure 
is [19,20] 
2
22eG T
h
=      (7) 
We assume a dimensionless strength for Delta function 
potential [21] by rescaling our 
parameters, 2 2/j j jmd U πΩ = h . In figure (1), we show 
the conductance through 5N =  Dirac delta function 
potential with strength 0.2Ω = . A perfect transmission 
in this case is in general impossible as predicted by [4, 
22]. According to reference [21] we can not have a 
resonant transmission, 1T = , even if N is very large.   
 
/kd π
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
G
 
Figure 1: Conductance spectrum G in the units of 22 /e h as a 
function of /kd π for a sequence of Dirac delta function 
potential with N=5. The strength of the potential here is 
0.2Ω = .  
 
 2.2 The Binomially Tailored Quantum Wire (BTQW) 
 
In this subsection, we reintroduce BTQW structure as 
shown in figure 2. The Dirac delta function has been 
equally spaced but their strength, jΩ , has been  
weighted according to the binomial distribution law, 
which is  
( ) / 2 , 0,1,....,Nj j
j
N
N N N
N
⎛ ⎞
Ω = =⎜ ⎟
⎝ ⎠
                  (8)  
 
 
 
( )jNΩ
jN
 
Figure 2: Shows binomially tailored Dirac delta function 
potential. jN values weighted by  equation (8). 
 
 
Thus, ( )jNΩ  represents the strength of the Dirac delta 
potential, 1N + represents the total number of Dirac delta 
function potentials in the quantum wire, and jN is the 
order of the Dirac delta potential. This novel structure of 
quantum wires can be released by putting metallic gates 
on top of a one dimensional electron gas and then by 
applying voltages, according to the binomial distribution 
law, to deplete the electron gas underneath.  In this case, 
equation (9) is no longer valid for our new structure. So 
that the total transmission matrix can be written as 
follows:  
 
  
( ) ( ) ( ) ( )
( ) ( ) ( )0........... 11
11
1
SikdS
ikdSikdSM NNNNt
⋅Γ⋅×
×Γ⋅⋅Γ⋅= ++
−
β
ββ
   (9) 
 
Notice that the potential strength is weighted according to 
equation (8).   In figure 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 ripples 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 is a forbidden band or conduction gap 
where the transmission is small.  
/kd π
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
G
 
Figure 3: Conductance spectrum G in the units of 22 /e h as a 
function of /kd π for a BTQW, with N=5. 
 
 
 
3. DEFECTS EFFECT ON THE ELECTRONIC 
CONDUCTANCE OF BTQW 
 
3.1 Strength Defect 
 
In this subsection, we study effect of strength defect of 
the central element of the binomial tailored quantum 
wire, and keeping the other elements and the spacing 
between the Dirac delta function potentials constant, on 
the electronic conductance through the BTQW. First, we 
consider the strength defect does not exceed 5%± of the 
Dirac delta function potential strength. That is, when the 
central Dirac delta functions potentials strength 
is ( ) ( )/ 2 1 0.05 / 2 1j jN NΩ + ± Ω + . In figure 4-a, we plot the 
electronic conductance spectrum for both strengths with 
35jN is and scaling factor of three. As can noticed there 
is slight difference between the two conductance 
spectrum curves, and defect free curves. In figure 4-b, we 
have increased the strength defect up to 20%± , we have 
noticed some measurable differences between the two the 
conductance spectrum curves and defect free curves, but 
still the conduction band and the forbidden bands well 
defined, which is a very good feature for BTQW.  
 
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
/kd π
G
 
Figure 4-a: The electronic conductance, in the units of 22 /e h as 
a function of /kd π . In this case, the defect is only 5%± , in the 
strength of the central Dirac delta function.  
 
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
/kd π
G
 
Figure 4-b: The electronic conductance, in the units of 22 /e h as 
a function of /kd π . In this case, the defect is only 20%± , in 
the strength of the central Dirac delta function.  
 
 
3.2 Dislocation Effect 
 
In this subsection, we study dislocation defect effect on 
the position of the central element in the BTQW, and 
keeping the other elements and the spacing between the 
Dirac delta function potentials constant. 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 
0.05d d± . In figure 5-a, we plot the electronic 
conductance spectrum for both dislocations with 
35jN is and scaling factor of three. Compared to defect 
curves, as can noticed there is a difference between the 
two curves. The conduction band starts lose its flatness 
and the forbidden band shaper for increased spacing 
between the central Dirac delta function and the adjacent 
one. In figure 5-b, we increase the dislocation defect up 
to 20%± , we have noticed measurable differences 
between the two curves and that of no defect case, but 
still the conduction band is well defined but the forbidden 
bands have a split compared to forbidden band in no 
defect curves. This splitting is due to resonant state in the 
forbidden energy band which leads to a bound state in the 
structure [3].  This is because the particle mode cannot 
propagate and hence get trapped.     
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
/kd π
G
 
Figure5-a: The electronic conductance, in the units of 22 /e h as 
a function of /kd π . In this case, the defect is only 5%± , in the 
position of the central Dirac delta function.  
/kd π
G
0.00
0.20
0.40
0.60
0.80
1.00
1.20
0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50 5.00
 
Figure 5-b: The electronic conductance, in the units of 22 /e h as 
a function of /kd π . In this case, the defect is only 20%± , in 
the position of the central Dirac delta function.  
 
4.  CONCLUSION 
 
We found the novel structure introduced by [13] has a 
good tolerance for strength reaches up to 20%±  and 
dislocation tolerance reaches up to 5%±  without losing 
the fascinating electronic transmission characteristics. 
 
 
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The deffect effect on electronic conductance in binomially tailored
  quantum wire

The deffect effect on electronic conductance in binomially tailored quantum wire

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