Physica E 47 (2013) 297–302

Contents lists available at SciVerse ScienceDirect

Physica E

1386-94

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/physe

Analytical and numerical study of quantum transport in an array ofnanorings: A case study with double rings

Farzad Khoeini a, Farhad Khoeini b,n

a Department of Electrical Engineering, Sharif University of Technology, P.O. Box 11365-9363, Tehran, Iranb Department of Physics, University of Zanjan, P.O. Box 45195-313, Zanjan, Iran

H I G H L I G H T S

G R A P H I C A L A

c We derive an analytical formula forelectron transmission across thesystem.

c We study the electron transport in aperfect double nanoring and designa NOR gate.

c Our approach is based on the tight-binding approximation and transfermatrix method.

c The threshold voltage is very sensi-tive to the changes of the magneticflux, coupling strength and ring size.

77/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.physe.2012.10.001

esponding author. Tel.: þ98 2415152648; fa

ail address: khoeini@znu.ac.ir (F. Khoeini).

B S T R A C T

We derive an analytical formula for electron transmission through an array of nanorings sandwichedbetween the two leads in different coupling strengths. Also, we design a NOR gate using the magneticfluxes inputs.

a r t i c l e i n f o

Article history:

Received 9 May 2012

Received in revised form

14 September 2012

Accepted 3 October 2012Available online 11 October 2012

a b s t r a c t

We present an analytical method to obtain an expression for electron transmission through an array of

disordered nanorings (ADNRs) sandwiched between two semi-infinite metallic leads in different lead-ring

coupling strengths. Our approach is based on the nearest-neighbor tight-binding approximation and

transfer matrix method. First, we derive an analytical formula for electron transmission across the system.

Next, we apply our approach to study of the electron transport in a perfect double nanoring (PDNR) and

design a NOR gate using the magnetic fluxes inputs. The conductance, current–voltage characteristics and

threshold voltage of the system are calculated in the weak and strong coupling regimes.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Although more than a decade has passed since the fabricationof the first nanorings (NRs) [1–3], these systems are amazing

ll rights reserved.

x: þ98 2415152264.

structures and have drawn an extensive attention because of theinterest in studying physical phenomena which take place in suchnanodevices such as the Aharonov–Bohm (AB) effect and theappearance of persistent currents [4–12]. Buttiker et al. [7] providedan early formulation of the persistent current in a mesoscopic metalring. Using a tight-binding model the two-terminal conductancewas also investigated by D’Amato et al. [13], and subsequently, byAldea et al. [14]. Aldea and co-authors studied the role of the width

F. Khoeini, F. Khoeini / Physica E 47 (2013) 297–302298

of the ring in producing the chaotic and fingerprint aspect of theaperiodic oscillations. A nontrivial change in the transport of an ABring is reported when the ring contains a quantum dot (QD) eitherembedded in an arm, or side coupled to it [15–17]. Recently, Janaet al. [18,19] investigated the role of the ring–dot and ring–ringcouplings in controlling the profile of transmission oscillations.Theoretical studies on double quantum rings coupled vertically andlaterally have been performed in Refs. [20–26]. The authors of Ref.[27] studied the effect of the proximity of the rings on thetransmission characteristics within a tight binding formulation.Cui et al. [28] investigated the electron transmission through astructure of serial mesoscopic metallic rings coupled to thetwo leads.

In our recent works, we have studied the electron transportproperties in nanodevices formed by nanoribbon and nanotubeunits in the absence of a magnetic field, numerically [29], while inthe present work, we concentrate on an analytical and numericalstudy of nanorings in the presence of magnetic field. The mainfocus of the paper is to obtain a general analytical formula forelectron transmission across an array of nanorings sandwichedbetween the electrodes, in different coupling regimes and tointroduce a NOR gate response. Simple mesoscopic rings withvoltage inputs can be used to fabricate all classical logic gates. OR,NOT, XOR, XNOR and NAND gates can be designed using only onering [30–34], while AND and NOR gates need to a double ring[25,26]. In this work, we introduce a NOR gate with magneticfluxes inputs in a two-ring system (see the first two rings inFig. 1(a)). The NOR gate behavior is specified by studying theconductance-energy and current–voltage (I–V) characteristics ofthe system. Our calculations are based on the tight-binding modeland transfer matrix method.

The paper is organized as follows. In Section 2, we introduce ourmodel and formalism, such as the tight-binding Hamiltonianfor the ADNR, and a renormalization method for mapping thesystem into a quantum wire. In Section 3, our numerical results forthe conductance and current are presented. Finally, we concludein Section 4 with a brief summary.

Fig. 1. (a) Schematic representation of an array of nanorings composed of M rings

attached to two semi-infinite metallic leads. The magnetic fluxes f1, f2, y, fM

are the inputs of the system. (b) Schematic view of a disordered quantum wire

after renormalization of the internal atoms of the rings (the renormalized version

of Fig. 1(a)).

2. Description of the model and renormalization method

In this section, we study the electronic quantum transport in anarray of disordered nanorings (ADNRs) formed of N¼N1þ

N2þ � � � þNM atoms connected to two semi-infinite metallicleads. The atoms of the leads are assumed comprised betweenð�1,0�

S½Nþ1,þ1Þ. The sandwiched spacer is known as an ADNR

device. First, we derive an analytical formula for electron transmis-sion across the ADNRs, in different coupling regimes. Next, weapply our approach to study the transport in a two-ring model. InFig. 1(a) we have depicted the structure of a symmetric ADNR. Thedevice size is denoted by N¼N1þN2þ � � � þNM , which representsthe total number of atoms, and Niði¼ 1,2, . . . ,MÞ is the number ofatoms in the ring i. In our study, we ignore the electron–phononand electron–electron interactions.

The model Hamiltonian is given by

H¼HLþHDLþHDþHDRþHR , ð1Þ

where HD, HL, and HR are the Hamiltonian of the device, the leftand the right leads, respectively. Also, HDLðDRÞ refers to theHamiltonian for the coupling between the device and the left(right) lead. The Hamiltonian of left (right) lead is defined as

HLðRÞ ¼X

j

e0LðRÞ9jS/j9þtLðRÞ

Xj

ð9jS/jþ19þ9jþ1S/j9Þ, ð2Þ

where tLðRÞ refers to the hopping energy between two atoms in theleft (right) lead in the nearest-neighbor sites approach. Also, inEq. (2), 9jS is the electron state at site j, and j varies betweenð�1,0� and ½Nþ1,þ1Þ. The ADNR is coupled to two leads by thecontact Hamiltonian, HDLðDRÞ given by

HDLðDRÞ ¼ tDLðDRÞð90ðNÞS/1ðNþ1Þ9þ91ðNþ1ÞS/0ðNÞ9Þ, ð3Þ

where tDLðDRÞ refers to the contact hopping energy between thedevice and left (right) lead. Also, the device Hamiltonian isgiven by

HD ¼HR1þHR1R2

þHR2þ � � � þHRM

, ð4Þ

where HR1ðR2Þ and HR1R2are the Hamiltonian of ring 1(2) and

coupling between the two rings 1 and 2, respectively. For the firsttwo rings the Hamiltonians are given by

HR1ðR2Þ ¼XN1ð2Þ

j ¼ 1

ej9jS/j9þX

j

ðvj,jþ1eiy1ð2Þ 9jS/jþ19þvjþ1,je�iy1ð2Þ 9jþ1S/j9Þ,

HR1R2¼ tR1R2

ð9N1=2þ1S/N1þ19þ9N1þ1S/N1=2þ19Þ: ð5Þ

where ej and vj,jþ1 are on-site energy and the hopping energybetween the nearest-neighbor atoms in the absence of magneticfield. For a pure system on-site energies and hopping integralshave constant values, while in a disordered system these energiesvary randomly. Due to the presence of magnetic flux f1ð2Þ

threaded by the ring 1(2), a phase factor y1ð2Þ ¼ 2pf1ð2Þ=N1ð2Þf0

appears in the hopping integrals of the rings (tj,jþ1 ¼ vj,jþ1e7 iy1ð2Þ ).Here N1ð2Þ represents the number of atomic sites in the ring 1(2).For the rings 3,4, . . . ,M the Hamiltonians are similar to the aboveequations. To calculate the conductance of the system using atransfer matrix technique [35–37], we map each ring ofthe system into a dimer by renormalization method [38,37]. Withfollowing approach, we obtain the renormalized parameters forevery disordered nanoring. The wave function at the site j isrelated to the wave functions at the sites j�1 and jþ1 by thedifference equation

ðE�ejÞcj ¼ tj,jþ1cjþ1þtj,j�1cj�1: ð6Þ

Also, the above equation for the atoms ðj�1Þ and ðjþ1Þ is asfollows:

ðE�ej71Þcj71 ¼ tj71,j72cj72þtj71,jcj: ð7Þ

F. Khoeini, F. Khoeini / Physica E 47 (2013) 297–302 299

By substituting cj from Eq. (6) into Eq. (7), and elimination of theatom j, the site energies the ðjþ1Þ th and ðj�1Þ th sites and alsothe hopping energies between these sites are modified as,

~ej71 ¼ ej71þtj71,jtj71,jtj,j71

E�ej,

~t j81,j71 ¼tj81,jtj,j71

E�ej: ð8Þ

The general recursive relations corresponding to the ðj�2Þ threnormalization, and then to the elimination of the site ðj�1Þ, arethe following:

~eðjÞ1 ðEÞ ¼ ~eðj�1Þ1 ðEÞþ ~t1,jþ1ðEÞ

1

E�~eðj�1Þjþ1 ðEÞ

~t jþ1,j,

~eðjÞjþ2ðEÞ ¼ ejþ2þtjþ1,jþ21

E�~eðj�1Þjþ1 ðEÞ

tjþ2,jþ1ðEÞ,

~t1,jþ2ðEÞ ¼ ~t1,jþ1ðEÞ1

E�~eðj�1Þjþ1 ðEÞ

tjþ1,jþ2, ð9Þ

and ~tn

jþ1,1 ¼~t1,jþ1 for n�2Z jZ1 (for upper/lower arm of each

single ring), the initial values being given by the Hamiltonianparameters, namely ~eð0Þj ðEÞ ¼ ej and ~t j,jþ1ðEÞ ¼ tj,jþ1.

We apply the above equations for the upper and lower arms M

disordered nanorings and obtain a disordered quantum wire(DQW) with M different dimers (with different renormalizedon-site energies and hopping integrals). As instance the renorma-lized parameters for the first ring of the array with N1 ¼ 8 atoms(for the dimer 1) are as

~e1D ¼ ~eð3Þupper1 ðEÞþ ~eð3Þlower

1 ðEÞ�e1,

~e2D ¼ ~eð3Þupper5 ðEÞþ ~eð3Þlower

5 ðEÞ�e5,

~v1,2 ¼ ~tupper1,5 ðEÞþ ~t

lower1,5 ðEÞ: ð10Þ

Now, we use the transfer matrix method to study the electrontransport through the DQW.

The time-independent Schrodinger equation H9cS¼ E9cS forthe DQW reads

clþ1 ¼E�~e lD

~vl,lþ1cl�

~vl,l�1

~vl,lþ1cl�1, ð11Þ

where clþ1 and ~e lD are the effective wave function of the DQW atthe site ðlþ1Þ and renormalized on-site energy at the site l,respectively. Here, l¼ 1, � � � ,N¼ 2M.

Eq. (11) can be rewritten via the 2�2 transfer matrix form as

clþ1

cl

!¼ Pl

cl

cl�1

!, ð12Þ

where Pl is given by

Pl ¼1

~vl,lþ1

E�~e lD � ~vl,l�1

~vl,lþ1 0

!: ð13Þ

From the above consideration, we can obtain a recurrent matrixequation for the whole system, that the wave functionsðcNþ2,cNþ1Þ can be related to ðc0,c�1Þ, as

cNþ2

cNþ1

!¼ PR � P � PL

c0

c�1

!: ð14Þ

Here, the total transfer matrix P¼Q1

l ¼ N Pl, creates a recurrentequation for the whole DQW with N sites, as

cNþ1

cN

!¼

P11 P12

P21 P22

!c1

c0

!: ð15Þ

The average transmission coefficient of the DQW is obtainedusing the method of Refs. [35,36] as

TNðE,VÞ ¼4t2

DR9sin yL sin yR9det2P

tLtR P11þtDLtL

� �P12eiyL�

tDRtR

� �P21eiyR�

tDLtL

� �tDRtR

� �P22eiðyR þyLÞ

��� ���2* +

:

ð16Þ

Here, / . . .S shows averaging over all possible realizations of theDQW. This formula is exact for a pure system and averaging is notnecessary. It is more general than the ones obtained in Refs.[39,28] because the parameter of tDLðDRÞ can be considered fordifferent ring–lead coupling strengths. Besides, it is applicable toboth diagonal (random on-site energies) and off-diagonal (ran-dom hopping energies) disorder in different ring sizes andmagnetic fluxes. At zero temperature and bias voltage the con-ductance of the ADNR is given by,

G¼2e2

hTNðEÞ: ð17Þ

The electrical current (I) that passes through the ADNR can beobtained from the following expression [40]:

IðVÞ ¼2e

h

Z EF þðeV=2Þ

EF�ðeV=2ÞTNðE,VÞ dE, ð18Þ

where EF and V are the equilibrium Fermi energy and applied biasvoltage, respectively.

In the next section, we are going to concentrate particularly ona pure two-ring model and show a NOR gate response withmagnetic flux inputs. Also, we obtain all the numerical results forthe zero temperature, however, they are valid even at finitetemperatures, because the broadening of the energy levels ofthe double ring due to its coupling to the leads is much largerthan that of the thermal broadening [40,26].

3. Numerical results and discussion

In the previous section, we obtained an analytical formula fortransmission probability, which is applicable to both symmetric(upper and lower arms of each ring contain equal number ofatoms) and asymmetric ring structures. In this section, we presentour numerical results for zero temperature conductance andcurrent flowing through the symmetric pure double nanoring,but all the results are valid even for finite temperatures. Thedevice size is set to N¼N1þN2 ¼ 16 atoms that is shown in Fig. 1(the first two rings), otherwise its value is expressed in thefigures, clearly. For simplicity, we choose c¼e¼h¼1 in ourpresent calculations. All atomic on-site energies in the leads andthe PDNR are set to zero. The hopping energy between thenearest-neighbor atoms of the PDNR is set at 3, while for theleads it is chosen as 4 [25,26]. Besides, the equilibrium Fermienergy EF is set at 0.

In this section, our numerical results are compared for thetwo distinct regimes of weak and strong couplings accordingto Refs. [25,26]. For the weak and strong coupling regimes wechoose tDLðDRÞ equal to 0.5 and 2.5, respectively. The key control-ling parameter for all these calculations are the magnetic fluxesf1 and f2, threaded by the rings 1 and 2, which are assumed to bebetween ½0,1�.

In Fig. 2, we have shown the conductance of a double nanoringas a function of the incident electron energy, in the weak couplinglimit (solid line) and strong one (dashed line) at several values ofthe magnetic fluxes. The curves have been plotted for the cases (a)f1 ¼f2 ¼ 0 and (b) f1 ¼ 0, f2 ¼ 0:5 and f1 ¼ 0:5, f2 ¼ 0 and f1 ¼

0:5, f2 ¼ 0:5. Fig. 2(a) shows that for the case f1 ¼f2 ¼ 0, i.e., thetwo inputs are low, the electron conduction through the system is

0.0

0.5

1.0

1.5

2.0

Weak coupling

Strong coupling

Energy, E- 8 - 6 - 4 - 2 0 2 4 6 8

Con

du

ctan

ce, G

-0.4

-0.2

0.0

0.2

0.4

φ1=0.5φ2=0.5φ1=0.0

φ1=0.5

φ2=0.5

φ2=0.0

φ2=0.0φ1=0.0

Fig. 2. Conductance G as a function of the incident energy E for a double nanoring

with N¼16 atoms in the weak coupling regime (solid line) and strong one (dashed

line). (a) f1 ¼f2 ¼ 0, (b) three cases f1 ¼ 0, f2 ¼ 0:5 and f1 ¼ 0:5, f2 ¼ 0 and

f1 ¼f2 ¼ 0:5.

Cu

rren

t, I

Weak coupling

Strong coupling

Voltage, V-8 -6 -4 -2 0 2 4 6 8

Cu

rren

t, I

-0.4

-0.2

0.0

0.2

0.4

-0.4

-0.2

0.0

0.2

0.4

-4

-2

0

2

4

-0.4

-0.2

0.0

0.2

0.4

Fig. 3. I–V characteristics for the double nanoring with N¼16 atoms in the weak

coupling regime (solid line) and strong one (dashed line). (a) f1 ¼f2 ¼ 0, (b) three

cases f1 ¼ 0, f2 ¼ 0:5 and f1 ¼ 0:5, f2 ¼ 0 and f1 ¼f2 ¼ 0:5.

F. Khoeini, F. Khoeini / Physica E 47 (2013) 297–302300

allowed. For some particular energies the conductance showssharp resonant peaks. At these resonant energies, the transmis-sion probability T goes to unity and therefore the conductanceapproaches the value 2 (G¼ 2TN). All these resonant peaks areassociated with the energy eigenvalues of the double nanoringand thus the conductance spectrum reveals the electronic struc-ture of the system. Also, we have investigated the effect of thecoupling strength of the double nanoring to the leads. In the caseof strong coupling (dashed line), all the resonant peaks get largewidths compared to the weak coupling limit (solid line).By tuning the coupling strength, we can get the electron trans-mission across the system for the wider range of energies.Fig. 2(b) shows that for the three cases f1 ¼ 0, f2 ¼ 0:5 andf1 ¼ 0:5, f2 ¼ 0 and f1 ¼f2 ¼ 0:5, i.e., when one of the twoinputs is high and another is low or the two inputs are high theconductance of the system becomes exactly zero. For a symmetricring threaded by an AB flux f, the probability amplitude ofgetting an electron across the ring becomes exactly zero for themagnetic flux, f¼ 0:5. This is because of the result of thequantum interference among the two waves in the two arms ofthe ring [41]. Therefore, for all these three cases the combinedeffect of these two rings provides vanishing transmission prob-ability across the PDNR. This feature clearly shows the NOR gatebehavior in the system. Our results are in good agreement with

those obtained using the Green’s function method in Refs. [25,42].For further investigation, we study the I–V characteristics of thesystem. The current passing through the double nanoring iscomputed from Eq. (18).

In Fig. 3, we show the variation of the current I as a function ofthe applied bias voltage V for the PDNR in the weak and strongcoupling regimes, where (a) f1 ¼f2 ¼ 0 and (b) f1 ¼ 0, f2 ¼ 0:5and f1 ¼ 0:5, f2 ¼ 0 and f1 ¼ 0:5, f2 ¼ 0:5. In Fig. 3(a), the non-zero current is observed when the magnetic fluxes are 0. In thestrong coupling regime, the current varies continuously with theapplied bias voltage and has a larger amplitude than the weakcoupling case. Also, in this regime, the system is almost metal(an ohmic regime at low bias voltage), since all the resonant peaksget broadened which provide large current in the integrationprocedure of the transmission function TN. Thus by tuning thestrength of the ring–lead coupling, we can achieve to a largecurrent amplitude from the low one for the same bias voltage V.For example at the bias voltage 6, the current is 4.19 for strongcoupling regime which is much larger compared to the weakcoupling case (0.17). From this figure it is observed that thecurrent passing through the system exhibits step-like behavior, inthe weak coupling case. This is due to the existence of the sharpresonant peaks in the conductance spectrum, since the current iscomputed from the transmission spectrum TN. With increasingthe bias voltage V, the chemical potentials on the electrodes areshifted and finally cross one of the quantized energy levels of thesystem. Therefore, a current channel is opened which produces ajump in the I–V profile. According to Fig. 3(b), for the cases ofeither the two inputs are set to 0.5 (f1 ¼f2 ¼ 0:5), or one of thetwo inputs is 0.5 and other is zero (f1 ¼ 0, f2 ¼ 0:5 or f1 ¼ 0:5,f2 ¼ 0), the current becomes exactly zero for the whole range ofthe bias voltage. The results could be clearly understood from theconductance spectra given in Fig. 2. From these I–V spectra, thebehavior of the NOR gate is clearly observed. This behavior islisted in Table 1, which gives values of the current amplitude atthe bias voltage V¼6, in the weak (strong) coupling limit. It showsI¼0.17 (4.19) only when the two inputs of the system are low(f1 ¼f2 ¼ 0), while for the other three cases when eitherf1 ¼f2 ¼ 0:5 or f1 ¼ 0, f2 ¼ 0:5 or f1 ¼ 0:5, f2 ¼ 0, the currentI gets the value 0. The present results show the NOR gate response

Table 1The truth table of the system represents NOR gate response in the weak (strong)

coupling regime. The output current I is computed at the bias voltage 6.

Input-I (f1) Input-II (f2) Current (I) in weak (strong)

coupling regime

0 0 0.17 (4.19)

0 0.5 0

0.5 0 0

0.5 0.5 0

Magnetic flux,

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

Strong coupling

Weak coupling

Ring size,

0.0 0. 1 0. 2 0. 3 0. 4

4 8 12 16 20

Th

resh

old

volt

age,

Vth

T

hre

shold

volt

age,

Vth

0.0

0.5

1.0

1.5

2.0

2.5

Strong coupling

Weak coupling

Fig. 4. Threshold voltage for the double nanoring in the two coupling regimes

with (a) N¼16 atoms and (b) f1 ¼f2 ¼ 0.

F. Khoeini, F. Khoeini / Physica E 47 (2013) 297–302 301

in the PDNR. Our results for the I–V characteristics are in goodagreement with those obtained in Refs. [25,42].

Addition to these behaviors, it is also important to note that thenon-zero value of the current appears beyond a finite value of V, theso-called threshold voltage (Vth). This Vth can be tuned by control-ling the magnetic flux and the size (N) of the two rings. In Fig. 4(a),we have shown the variation of the threshold voltage as a functionof the magnetic flux in the two different regimes. In the weakcoupling regime, with increasing the magnetic flux, the thresholdvoltage rises, linearly, but in the strong one, this increase is almostexponentially. Also, in Fig. 4(b) we have illustrated the variation ofthe threshold voltage as a function of the device size for the tworegimes. In the weak coupling limit, with increasing the ring size,the threshold voltage reduces almost exponentially, but it is fixed inthe strong coupling limit. The calculated results, therefore, show

that the conductance, current, and threshold voltage are sensitive tothe parameters such as the magnetic flux and ring size.

4. Conclusions

Based on the tight-binding model and the renormalizationapproach, we mapped a series of disordered nanorings into adisordered quantum wire to obtain an analytical formula for electrontransmission across the system by the transfer matrix method indifferent coupling regimes. The formulation has also been tested forthe pure double nanoring with the parameters introduced in Ref.[25], at the zero temperature. However, all the results are valid evenfor finite temperatures. Because the broadening of the energy levelsof the device due to the device–lead coupling is much larger than thatof the thermal broadening [40,26]. We determined the conductance-energy as well as current–voltage characteristics of the system indifferent magnetic fluxes and the coupling regimes. Our numericalresults for the I–V characteristics and the conductance spectrumshow the NOR gate response. In addition, we found the thresholdvoltage is very sensitive to the changes of the parameters such as themagnetic flux, ring–lead coupling strength and ring size (namely, it isalmost independent of the ring size in the strong coupling regime,while decreases for larger sizes in the weak coupling limit, and growswith increasing the magnetic flux). Our results might have significantapplications in controlling the electron transport in nanodevices.

References

[1] J.M. Garcıa, G. Medeiros-Ribeiro, K. Schmidt, T. Ngo, J.L. Feng, A. Lorke,J. Kotthaus, P.M. Petroff, Applied Physics Letters 71 (1997) 2014.

[2] A. Lorke, R.J. Luyken, Physica B 256–258 (1998) 424.[3] A. Lorke, R.J. Luyken, A.O. Govorov, J.P. Kotthaus, J.M. Garcia, P.M. Petroff,

Physical Review Letters 84 (2000) 2223.[4] Y. Aharonov, D. Bohm, Physical Review 115 (1959) 485.[5] T. Chakraborty, P. Pietilainen, Physical Review B 50 (1994) 8460.[6] S. Viefers, P. Koskinen, P.S. Deo, M. Manninen, Physica E 21 (2004) 1.[7] M. Buttiker, Y. Imry, R. Landauer, Physics Letters A 96 (1983) 365.[8] N.A.J.M. Kleemans, I.M.A. Bominaar-Silkens, V.M. Fomin, V.N. Gladilin,

D. Granados, A.G. Taboada, J.M. Garcıa, P. Offermans, U. Zeitler,P.C.M. Christianen, J.C. Maan, J.T. Devreese, P.M. Koenraad, Physical Review Letters99 (2007) 146808.

[9] J.M. Escartın, M. Barranco, M. Pi, Physical Review B 82 (2010) 195427.[10] E. Faizabadi, M. Omidi, Physics Letters A 374 (2010) 1762.[11] J. Planelles, J.I. Climente, European Physical Journal B 48 (2005) 65.[12] L. Zhang, F. Liang, J. Wang, European Physical Journal B 74 (2010) 91.[13] J.L. D’Amato, H.M. Pastawski, J.F. Weisz, Physical Review B 39 (1989) 3554.[14] A. Aldea, P. Gartner, I. Corcotoi, Physical Review B 45 (1992) 14122.[15] A. Yacoby, M. Heiblum, D. Mahalu, H. Shtrikman, Physical Review Letters 74

(1995) 4047.[16] R. Schuster, E. Buks, M. Heiblum, D. Mahalu, V. Umansky, H. Shtrikman,

Nature 385 (1997) 417.[17] A.W. Holleitner, C.R. Decker, H. Qin, K. Eberl, R.H. Blick, Physical Review

Letters 87 (2001) 256802.[18] S. Jana, A. Chakrabarti, Physical Review B 77 (2008) 155310.[19] S. Jana, A. Chakrabarti, Physica Status Solidi B 248 (2011) 725.[20] L.G.G.V. Dias da Silva, J.M. Villas-Boas, S.E. Ulloa, Physical Review B 76 (2007)

155306.[21] M. Royo, F. Malet, M. Barranco, M. Pi, J. Planelles, Physical Review B 78 (2008)

165308.[22] L.K. Castelano, G.-Q. Hai, B. Partoens, F.M. Peeters, Physical Review B 78

(2008) 195315.[23] L.K. Castelano, G.-Q. Hai, B. Partoens, F.M. Peeters, Journal of Applied Physics

106 (2009) 073702.[24] T. Chwiej, B. Szafran, Physical Review B 78 (2008) 245306.[25] S.K. Maiti, Physics Letters A 373 (2009) 4470.[26] S.K. Maiti, Solid State Communications 149 (2009) 2146.[27] S. Jana, N. Bhattacharya, A. Chakrabarti, Physica B 406 (2011) 4387.[28] W.Y. Cui, S.Z. Wu, G. Jina, X. Zhaob, Y.Q. Ma, European Physical Journal B 59

(2007) 47.[29] Farhad Khoeini, A.A. Shokri, Farzad Khoeini, European Physical Journal B 75

(2010) 505;F. Khoeini, A.A. Shokri, Journal of Computational and Theoretical Nanoscience8 (2011) 740;F. Khoeini, A.A. Shokri, Journal of Computational and Theoretical Nanoscience8 (2011) 1315;A.A. Shokri, F. Khoeini, Superlattices and Microstructures 51 (2012) 785.

F. Khoeini, F. Khoeini / Physica E 47 (2013) 297–302302

[30] S.K. Maiti, Solid State Communications 149 (2009) 1684.[31] S.K. Maiti, Journal of Computational and Theoretical Nanoscience 7 (2010)

594.[32] S.K. Maiti, Solid State Communications 149 (2009) 1623.[33] S.K. Maiti, Journal of the Physical Society of Japan 78 (2009) 114602.[34] S.K. Maiti, Physica Scripta 80 (2009) 055704.[35] M. Mardaani, A.A. Shokri, K. Esfarjani, Physica E 28 (2005) 150;

A.A. Shokri, M. Mardaani, K. Esfarjani, Physica E 27 (2005) 325.[36] F. Khoeini, A.A. Shokri, H. Farman, Physica E 41 (2009) 1533.

[37] A.A. Shokri, F. Khoeini, European Physical Journal B 78 (2010) 59.[38] R. Farchioni, G. Grosso, G.P. Parravicini, Physical Review B 53 (1996) 4294.[39] A. Chakrabarti, R.A. Romer, M. Schreiber, Physical Review B 68 (2003)

195417.[40] S. Datta, Electronic Transport in Mesoscopic Systems, Cambridge University

Press, Cambridge, 1997.[41] R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Physical Review Letters

54 (1985) 2696.[42] S.K. Maiti, Physica E 36 (2007) 199.