 reached a stage where devices can be fabricated at sizes smaller than the single particle electronic coherence length. In such mesoscopic systems the wavefunction maintains its phase coherence so that the electron can travel coherently through the sample. The access to "coherent transport" is granted by the advances made in lithography techniques which have opened a very rich field of theoretical and experimental research concerning quantum wires and quantum dots. Electro-optical experiments in solid state devices could lead to new switches which use the quantum wave nature of the electron.
Many interesting quantum effects can be also found in coupled nanostructures where the electronic transport is drastically affected by the phenomenom of quantum interference.
Moreover, the application of a magnetic field, which is often used to probe the properties of devices, can also induce characteristic changes in the phase coherence of the electronic wavefunctions 3 which, in turn, give rise to particular interference effects for the electronic transport. In the pioneering work of Aharonov and Bohm 4 such an effect was demonstrated via a thought experiment and it was shown that the conductance of a ring should oscillate as a function of the magnetic flux threaded through it. Among the manifestations of the Aharonov Bohm (AB) effect [5][6][7][8][9] usually are the periodic magnetoresistance oscillations in normal metal rings and in electrostatically defined heterjunction rings. The AB effect is a result of the relative phase shift between two electron beams enclosing a magnetic flux φ, where the magnetic field causes a 2π φ φ 0 change of the phase difference between the two arms of the ring. In this system the magnetoresistance oscillations have period φ 0, which allows tuning of the phase of a wavepacket with destructive and constructive interference in cycles.
Owing to the great variety of the possible configurations for quantum dots it is of great interest to investigate the change of the interference effect in AB rings when the channel number is greater than two. In this paper, we study the electron transport properties of such a multi-chain structure with common leads attached at its ends. We show that it can provide many alternative options for tuning quantum interference effects in the electronic transport.
In the multi-channel structure an initial wave splits up into complementary waves ψ 1... ψ N, where N is the total number of chains involved. These waves propagate independently in every chain and are finally recombined at the outgoing lead. Interference effects among the different waves can be observed from the behavior of the electrical resistance calculated between the two leads. We also show that if the number of chains involved in the system is large enough most of the states are reflected and only a few of them can propagate through the system. This kind of "blocking" or "localization" of the electron waves, which occurs via quantum interference effects caused by the geometrical structure, is observed despite the absence of any disorder in the system.
In the presence of a magnetic field an electron moving around a loop will experience a phase change determined by the flux threaded through the loop. In a multi-chain system the phase changes are not the same for different paths of propagation so that they can lead to particular interference phenomena accompanied by much more complicated conductance oscillations than in the ordinary two-channel AB ring. We find that the pattern of these magnetoquantum oscillations is very sensitive to the number and the length distribution of the chains involved in the structure. Moreover, the oscillation patterns and their periodicities are also very sensitive to the partitioning of the flux among the areas enclosed by the paths.
It should be pointed out that the results obtained in this paper could be useful towards understanding quantum dots with a special configuration.
The structure of the paper with the exposition of our results is as follows: in section II we describe the studied structure and give analytic expressions for the electronic transport, in section III we demonstrate different kinds of transport induced by special quantum interference effects with and without a magnetic field. The obtained results are summarized and discussed in Section IV.

II. MODEL AND FORMULA
We consider a ring of many chains with two common leads at its ends, threaded by a magnetic field which produces a flux in every loop enclosed by two nearest-neighbor chains.
In addition we suppose that the multi-chain system is embedded in an infinite perfectly conducting chain with a left and a right part serving as the two electrodes. The configuration is shown in Fig.1 and the transport properties for non-interacting electrons in this system are studied via the tight binding Hamiltonian where Our picture of the electronic transport in the system consists of an electron wave incident from the source into the perfect chain, then ramified into the N chains also experiencing different phase increments and eventually recombined into one channel at the output lead.
Thus, an electronic beam incident from the right should be partially transmitted and partially reflected by the multi-chain system. In the site representation the coefficients of the wave function at the left and the right parts of the pure chain can be written as where k = cos −1 (E/2) is the wave vector of a wave function with energy E, A is the amplitude of the incident wave, R is the amplitude of the reflected wave and the wave function for all the bonds is normalized so that the transmitted wave amplitude is unity.
The transmission coefficient which measures the transparancy of the system is, subsequently, defined as |t| 2 = 1/|A| 2. The wave function coefficients in the αth chain, by including the left node 0, can be expressed as a linear combination of the propagating and the reflected plane waves via where a α,j is the coefficient at the jth site of the αth chain. The coefficient at the left lead node j=0 from Eqs.
(2) and (3) gives the relation Similarly, we can calculate the coefficient at the right node j=s, by including the flux-induced phase shift term, and a comparison with Eq.
(2) gives a second relation Therefore, from the two Eqs. (4) and (5) we can express the wave function coefficient of Eq.
(3) for all chains α via in terms of A and R.
On the other hand from the Schrödinger difference equations at the two lead nodes 0 and s we obtain and by a substitution of Eqs. (6), (7) into Eqs. (8), (9) eventually obtain two equations for the A and R. Finally, the transmission coefficient can be calculated from their solution, which gives where for arbitrary chain numbers N α, α = 1, 2,..., N.
Eq. (10) is the most important result of this paper, which presents the general analytical expression for the transmission coefficient in a multi-channel ring. This expression can be further simplified for special geometries, for example, if the chain lengths are equal to N 1 = N 2 =... = N N ≡ L, and the two nearest neighbor chains enclose a fixed area and a simpler exact expression for the transmission coefficient |t(E)| 2 can be obtained. It can be seen from Eq. (10) that the magnetic field dependence is solely due to |f 0 | so that if the factors 1 at a Fermi energy E.

CONFIGURATIONS
In this section we show our results associated with quantum interference effects in multichain systems by application of Eq. (10). Our purpose is to illustrate the relationship between electronic wave transport in the multi-channel system and the geometric ring structure, also in the presence of an external magnetic field.

A. Equal chain system without a magnetic field
In the absence of a magnetic field the replacement φ = 0 for f 0, c 0 is made in Eqs.
(11), (12). The obtained results in this case correspond to a similar model of a total number of N thin wires joined together at their two ends as it was introduced by Wang et al 11 in their study of electronic transport through a quantum cavity. These authors have predicted that the total electron transmission can be simply expressed as a coherent sum of the transmission coefficients obtained from every chain. Our results can give an even more complicated transmission behavior due to the geometrical structure of the electrodes.
In Fig.2 we plot the transmission coefficient versus the electronic energy for such a multichain system made of equal chains. The pattern shown exhibits an interesting bridge-arc shape whose curvature becomes larger and the blank region below the arc smaller if the chain number increases, with some states still having high values of the transmission coefficient.
Thus, if the chain number is large enough most of the states are reflected and only very few states can propagate through the multi-channel system. It must be emphasised that the blocking of the electron propagation at most energies is merely caused by quantum interference due to the geometrical structure involved. It seems, however, a puzzle why such a rather symmetric geometry can give rise to a very complicated behavior of the outgoing wave. This is probably due to the fact that the translation symmetry is broken at the two contacts, which leads to partial destructive interference of the electron waves. Wang et al 11 have also observed a partial blocking of electron waves in the propagation regime of a quantum-wave-filter consisting of field-induced nanoscale cavities and 1D wires by varying the electronic wavelength. Our results could account for the reported experimental behavior.
We have also investigated the relationship between the obtained features of the conducting spectrum and the chain number for an equal chain multi-chain system. In Fig. 3 the conduction band as a function of the number of chains is illustrated and we observe that by increasing the chain number the transmission pattern becomes more and more sparse and the conduction band splits into several subbands. If the number of chains becomes large enough we find that most of the states cannot propagate through the system, becoming "blocked" or "localized". Thus, a quantum interference effect causes the conduction band to become discrete in the absence of disorder and/or interchain couplings, except at the connections of the system at the two end nodes. A relation between the conductance and the chain number can be extracted