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Surface defects characterization in quantum wires by acoustical phonons scattering M.S. Rabia

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Laboratoire de Me´canique des Structures et Energe´tique, De´partement de Ge´nie Me´canique, Faculte´ du Ge´nie de la Construction, Universite´ M. Mammeri, Tizi-Ouzou 15000, Algeria Received 11 July 2006; accepted 1 August 2006 Available online 17 August 2006

Abstract We investigate, in the harmonic approximation, the inﬂuence of local defects on scattering properties of elastic waves in perturbed quasi-one-dimensional planar waveguide. Our model is based on two inﬁnite quantum wires in which diﬀerent adsorbed scatterers (or surface defects) are considered. Following the approach of Landauer, we solve the dynamical equation using scattering boundary conditions and applying the matching method formalism. A detailed study of the defect-induced ﬂuctuations in the transmission and conductance spectra is presented for the scattering process for two defect conﬁgurations. As in the electronic case, the presence of localized states in the frequency range of propagating states leads to additional structures which can be related with Fano type resonant states and Fabry-Pe´rot oscillations. The Fano resonance shifts to lower energy for increasing defect mass. Numerical results reveal the intimate relation between transmission spectra and localized impurity states and provide a basis for the understanding of conductance spectroscopy experiments in mesoscopic systems. 2006 Elsevier B.V. All rights reserved. Keywords: Mesoscopic disordered systems; Phonons scattering; Defect in nanostructures; Matching method

1. Introduction During these last twenty years, most interference eﬀects in the properties of transport was established as well in an experimental way [1] that theoretical [2]. Basis of coherent transport, known as quantum, and their generalization to multichannel systems were introduced by Landauer [3] who related the conductance of the system to its scattering matrix. Most of the recent works [4–6] relate to the study of the electronic diﬀusion in quasi one-dimensional systems. Following the example study undertaken by Garcia-Moliner et al. [7] applied to the phonon scattering theory, these works generally call upon the Green function approach. The basic motivation being the comprehension of the limitations which nano-

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0166-1280/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2006.08.032

structural disorder can induce on the physical properties of low-dimensional samples. The practical applications are numerous as well in traditional metallurgy, electrochemistry, catalysis as in electronics. More recently, several authors [8–11] showed that multiple scattering introducing the interference eﬀects occupies a privileged place in the description of the transport phenomena. Elastic scattering is an essential tool in the investigation of the structural properties of materials [12–15], within the wavelength limit. The X-rays, for example, which are largely used in the analysis of the structures, have their limit on a nanometric scale. Here, the vibratory waves can lead to an interesting alternative, at least whenever the probabilities of inelastic scattering due to the electron–phonon diﬀusions and phonon–phonon can be kept suﬃciently small. In this work, we present a study relating to the diﬀusion of phonons in an inﬁnite crystal comprising a defect of structure. We analyze the behaviour of a vibratory wave

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being propagated in a double linear chain, which is assimilated to a quasi one-dimensional perfect waveguide in which a scatterer (or defect) is physically adsorbed on surface (adatom). We are interested with the reﬂected and transmitted parts of incidental wave, atom displacements in the perturbed area and their evolution relatively to the adatom mass and the bonding constants strength with the network. The mathematical treatment of the problem resorts to the matching method [6,16] within the harmonic approximation framework [17] by using scattering boundary conditions. The outline of this paper is organized as follows. Section 2 refers to the description of the studied model. After a brief summary of the method principle, we calculate the dynamical properties of the planar quasi one-dimensional waveguide and introduce the mathematical formalism used to treat the scattering in presence of defects in Section 3. The numerical results analysis and discussion will be the subject of Section 4. Section 5 will be devoted to the conclusion.

3. Matching method principle The matching method, initiated by Feuchtwang in 1967 then developed by Szeftel and Khater in 1987, oﬀers unquestionable advantages: not only it returns account in a satisfactory way of the phonon dispersion curves [18] and of surface resonances, but it gives also a more general deﬁnition of the concept of resonance and allows a more transparent analysis of the behaviour of displacements in the vicinity of Van Hove singularities. Its execution requires the subdivision of the crystal in three distinct areas, having all the same two-dimensional periodicity along surface: (i) A bulk area of two-dimensional periodicity representing the perfect crystal without any defect (perfect waveguide) where the dispersion curves are initially derived. It is comparable with the areas L and R of Fig. 1. (ii) A perturbed region consisting of an arbitrary number of adsorbates, reconstructed or relaxed layers inside which the translational symmetry is lost along that direction not contained in the surface plan. It is represented by the M region of Fig. 1. (iii) An intermediate region of bulk matter, the thickness of which increases with the increasing range of interlayer interactions, used to match bulk phonons with the boundary conditions imposed by the surface; from where the technique was given the name of ‘‘matching method’’. In our case, this area is represented on both sides of M region by the matching areas which make it possible to relate the defect atomic displacements to those of the two L and R semi-inﬁnite perfect waveguides (Fig. 1).

2. Model It consists of two linear parallel periodic chains of masses, assimilated to a quasi-one-dimensional plane waveguide in which are incorporated scatterers (or defects) on the surface. The parallel chains are composed of speciﬁc masses aligned along the direction of propagation (x axis). The situation is depicted in Fig. 1. Each mass is linked to its nearest and next nearest neighbours by harmonic springs of stiﬀness constants k1 and k2. To simplify, the distances between adjacent masses are considered equal in the two Cartesian directions x and y of the plan. Also, to take account of the modiﬁcation of the bonding strength ﬁeld in the interacting region M, we introduce a proportionality factor k which indicates the ratio of the diﬀerent force constants between the defect zone masses and those of the perfect lattice areas L (left) and R (right-hand side) located in sites separated by equivalent distances.

3.1. Dynamics of the perfect waveguide 3.1.1. Propagating modes Within the harmonic approximation framework [17], the dynamics of the perfect waveguide is described by the clas-

y Reflected wave

α klv

k2

Incidental wave

Klv

Transmitted wave

kv

k1

x

……

-2 L

-1

k1

0

kl

1

M

2

3

…… R

Matching Regions

Fig. 1. Schematic representation of a planar quasi-1D waveguide made up of two linear inﬁnite chains perturbed by a surface defect. The grey area M indicates defect region, L and R two semi-inﬁnite perfect waveguides.

M.S. Rabia / Journal of Molecular Structure: THEOCHEM 777 (2006) 131–138

sical Newton’s equations. The corresponding equation of motion of an atom occupying the site (i, j) deﬁnes the displacement amplitudes uija in the direction a, where a = x, y. For the reducible atom of the double quantum chain, projection on the Cartesian axes x(y) yields: mx2 uijx ¼ k 1 ðuijx uiþ1;j;x Þ k 1 ðuijx ui1;j;x Þ k2 k2 ðuijx uiþ1;jþ1;x Þ ðuijx ui1;jþ1;x Þ 2 2 k2 k2 ðuijx uiþ1;j1;x Þ ðuijx ui1;j1;x Þ 2 2 k2 k2 ðuijy uiþ1;jþ1;y Þ þ ðuijy ui1;jþ1;y Þ 2 2 k2 k2 þ ðuijy uiþ1;j1;y Þ ðuijy ui1;j1;y Þ 2 2 ð1aÞ k2 k2 ðuijx uiþ1;jþ1;y Þ þ ðuijx ui1;jþ1;y Þ 2 2 k2 k2 þ ðuijx uiþ1;j1;x Þ ðuijx ui1;j1;x Þ 2 2 k 1 ðuijy ui;jþ1;y Þ k 1 ðuijy ui;j1;y Þ k2 k2 ðuijy uiþ1;jþ1;y Þ ðuijy ui1;jþ1;y Þ 2 2 k2 k2 ðuijy uiþ1;j1;x Þ ðuijy ui1;j1;x Þ 2 2

mx2 uijy ¼

ð1bÞ where x is the vibration frequency. The corresponding dynamical equations for boundary sites (i ± 1, j ± 1) are obtained by setting terms corresponding to absent springs to zero. Note that the diagonal springs that couple the displacements parallel and perpendicular to the x axis are necessary to stabilize the system with respect to shear. Eq. (1a), (1b) must be solved by taking account of the boundary conditions for which we obtain the plane-waves solutions ~ ui ¼ ~ u0 eiqxi

ð2Þ

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2

is the normalized dimensionless frequency, where X2 ¼ mx k1 D(r2, Z) the dynamical matrix (4 · 4) of the perfect network (area L or R of Fig. 1). It contains terms in Z and 1 . The dimensionless quantity r2 ¼ kk21 denoting the bonding Z force constants ratio will be convenient for later purposes. The resolution of Eq. (3) for a ﬁxed value Z ¼ eiq makes it possible to obtain the eigenvalue vibration frequencies Xm as well as the associated eigenvectors ~ um . The propagating modes correspond to the solutions jZj = 1. They are usually given in terms of q, with q running over the ﬁrst Brillouin zone [p p]. In the case of the double chain, we obtain two acoustic modes with X ﬁ 0 when q ﬁ 0. The two remaining modes are optical with X „ 0 for any q. Fig. 2 shows the curves of dispersion X(q) for k1 = 1, r2 = 0.5, and m = 1. The analysis of the eigenvectors indicates that the eigenmodes of the waveguide are either symmetric or antisymmetric relatively to the central axis confused with the direction of propagation x. The results show that there is one acoustical and one optical mode for each symmetry. Moreover, the anticrossing behaviour between the symmetric acoustical and optical modes observed in Fig. 2 is due to the fact that the dispersion curves belonging to the same symmetry interact and therefore do not cross. Note further that the antisymmetric transverse acoustic mode has a q2 dispersion for q ﬁ 0 [19]. This behaviour is a consequence of the ﬁnite extension of the waveguide in y direction. 3.1.2. Evanescent modes In addition to the propagating modes calculated previously, the scattering in presence of defect requires also the determination of the evanescent modes (jZj < 1) of the system. These solutions can be obtained by diﬀerent procedures [6,20]. Method used here [14] consists of the increase of the system eigenvectors basis: 1 mija ¼ uija Z

ð4Þ

where q is the real wave vector, xi denotes the equilibrium position of column i, and the vector ~ ui describes the displacement amplitudes of each atom in this column 0 1 ui1x Bu C B i1y C ui ¼ B C @ ui2x A ui2y In prevention of the scatterer, it would be convenient to relate the displacement vectors of two adjacent columns by a phase factor Z (or attenuation factor) such that ~ uiþ1 ¼ Z~ ui . This relation is an essential feature of the matching method [6,16], initially used for the study of the localized phonons and surface resonances. For itinerant waves of the Eq. (2), we have Z ¼ eiqa ¼ eiq by taking a = 1. The eigenvalue problem of Eq. (1a), (1b) can then be written as Dðr2 ; ZÞ ~ ui ¼ X2~ ui

ð3Þ

Fig. 2. Dispersion branches for propagating modes of phonons for a quasi one-dimensional waveguide represented by an inﬁnite double quantum chain with q running over the ﬁrst Brillouin zone with k1 = 1, k2 = 0.5, and m = 1.

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We then rewrite the Eq. (2) in the form of an eigenvalue problem for Z, i ~ u ~ ~ ~ ð5Þ AðXÞ:W ¼ ZB W with W ¼ ~ mi Where A and B are (8 · 8) matrices coming from the basis change. Note that the dimension of this generalized eigenvalue problem is twice as large as the original problem. The resolution of the Eq. (5) yields pairs of eigenvalues Zm and Z 1 m which must be sort. The propagating solutions jZj = 1, described by real wave vectors, can be grouped in pairs corresponding to the two directions of propagation. Owing to the fact that they contain same information, we arbitrarily choose those which are propagated from the left to the right. The solutions jZj „ 1 correspond to evanescent or divergent waves. Only the physical relevant evanescent modes, necessary for the description of the scattering in the presence of defects, are retained. On Fig. 3 are represented the functional behaviours of the four vibrating modes characterizing the double chain. To facilitate the comparison between curves of dispersion X(q) and the curves X(Z), we knowingly placed Figs. 2 and 3 one at the top of the other by marking the common special points in the two representations. The projection of the curves on the complex plan shows that the propagating solutions follow the circles of unit radius, equal to the phase factor module; whereas the evanescent solutions correspond to the curves contained inside the unit circle. The two antisymmetric modes are degenerated at the point a (Z = 1, X = 0). The acoustic mode is propagated

up to the point f where X = Xf and Z = 1. It becomes evanescent for larger frequencies. Starting from point a, the other solution is immediately evanescent at the low frequencies then follows the optical branch of discrepancy between points h (Z = 1) and k (Z = 1) to become evanescent again. For still higher frequencies X > Xk, the two solutions follow the real negative Z axis and remain evanescent when jZj ﬁ 0 and X ﬁ 1. For the symmetric modes, the functional behaviour is somewhat more complicated. This is essentially due to the anticrossing phenomenon of the propagating acoustical and optical modes of the same symmetry, which constrained phonons to take evanescent paths to jump from a propagating branch to the other in the interacting zone, surrounded by a dashed circle in Fig. 2 [12]. The solution starting at point a follows the propagating acoustical mode up to point c, which corresponds to the maximum frequency in this branch. It then joins the minimum of the optical branch by taking an evanescent path with jZj „ 1. From point d to e, it continues on the propagating optical branch, before becoming again evanescent with real negative Z. The second solution is evanescent for X = 0, starting with a real negative value of Z. For increasing frequencies, it follows the negative Z axis to reach point f in the propagating optical branch, continues on this propagating branch up to its maximum frequency at point c 0 and then joins point d 0 on the propagating optical branch via an evanescent path. It coincides with the propagating optical branch between points d 0 and g and then becomes evanescent with real positive Z. For higher frequencies,

Fig. 3. Functional behaviour X (Z) of the vibrating modes in the double quantum chain. For comparison, the dispersion curves X (q) for the corresponding propagating modes are shown just on top.

M.S. Rabia / Journal of Molecular Structure: THEOCHEM 777 (2006) 131–138

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both solutions remain evanescent with jZj ﬁ 0 when X ﬁ 1.

where the sum is carried out over all propagating modes at the same frequency X.

3.2. Elastic scattering at defects

4. Numerical results

Since the perfect waveguides do not couple between different vibrational eigenmodes, we can treat the scattering problem for each eigenmode separately. For an incoming wave from the left of Fig. 1 in mode m,

4.1. Single adatomic scatterer

V~iin ¼ ðZ m Þi~ um ;

i 6 1

ð6Þ

where Z m is the attenuation factor of the entering mode, ~ um its eigenvector; i indicates the site occupied by the atom with respect to the direction of propagation. The resulting scattered waves are composed of a reﬂected and transmitted parts, which can be expressed as a superposition of the eigenmodes of the perfect waveguide at the same frequency, i.e., X 1 i 1 i ~ ur ¼ rmm : :~ um ; i 6 1 ð7Þ Zm Zm m X ~ uit ¼ tmm :½Z m i :~ um ðZ m Þ; i P 2 ð8Þ m

where rmm and tmm indicates the reﬂection and transmission coeﬃcients normalized beforehand by group velocities (slopes of the dispersion curves in Fig. 2) of the plane wave, set equal to zero for the evanescent modes. The evanescent modes are needed for a complete description of scattering in presence of defect, although they do not contribute at all to the energy transport. With the deﬁnitions (7) and (8), we can rewrite the dynamical equations for the perturbed double chain. Since there are perfect waveguides in regions L and R, we only need to solve Eq. (1a), (1b) for the masses inside the perturbed zone M and in the boundary columns (1) and (2), which are matched to the rest of the perfect waveguide by Eqs. (7) and (8). This yields a linear system of equations with eighteen unknowns, namely, the ten displacement amplitudes uija of the perturbed region M and the eight transmission and reﬂection tmm and rmm coeﬃcients. Isolating the inhomogeneous terms describing the incidental wave, we obtain an inhomogeneous system of linear equations ~ ¼ ½Df ðX; r2 ; ZÞV~in ½Df ðX; r2 ; k; ZÞ½RX

ð9Þ

where Df(X, r2, k, Z) indicates the dynamical defect matrix, ~ the vector gathering all the problem unknowns, V ~in the X incidental vector and R the matching matrix. The overall transmission (or conductance) of mesoscopic disordered multichannel systems at a given frequency X is signiﬁcant for calculating experimentally measurable physical quantities. It is then useful to deﬁne the total transmission K by summing over all input and output channels, X KðXÞ ¼ tmm ð10Þ m;m

The simplest defect conﬁguration to study is the double chain containing a single adatomic impurity. Phonons scattered by the adsorbate adatom are analysed relatively to an incidental wave coming from the left to the right (Fig. 1) with unit amplitude and a null dephasing on the boundary column of atoms (1). We thus consider the case where the propagating wave is done in x direction. Since both symmetric and antisymmetric modes show qualitatively the same behaviour, we present only the transmission spectrum of the antisymmetric modes. Fig. 4 shows the transmission coeﬃcients as a function of the dimensionless frequency X for light (m 0 = 0.3 m) and heavy (m 0 = 3 m) defect masses when the dimensionless physical parameters are r2 = 0.5, k = 0.85. The presence of defect leads to a general decrease of the transmission probability amplitude. As expected, the inﬂuence of the defect in acoustical regime is relatively weak, for X ﬁ 0 we get t11 ﬁ 1 independent of the perturbating mass (Fig. 4a). The subscripts on tmm refer to dispersion curves of Fig. 2, where the modes are numbered 1–4 from bottom to top. It is noted that mode 2, however propagating on this frequency range, is not excited; the two eigenmodes being perpendicularly polarized do not interfere (see Fig. 4). Moreover, the wave is quasi-transmitted at weak frequency and the reﬂection coeﬃcient increases gradually. Namely, backscattering becomes more important for q wave vectors near the zone boundaries, where t11, t33 ﬁ 0 independent of the strength of the defect. The conservation of energy may be used to show that jrmm j þ jtmm j ¼ 1, so rmm

Fig. 4. Transmission probabilities in the antisymmetric modes as a function of dimensionless frequency in a double chain containing single adatomic impurity. Full line, heavy defect (m 0 = 3m); dotted line, light defect (m0 ¼ m3 ).

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follows immediately. This relation will be used systematically as a check for all results along this work. At low frequencies, the energy of the network atoms is insuﬃcient to excite the defect atoms, all occurs as if the defect did not exist. Thus the incidental wave is found completely transmitted in this frequency range. In other words, the defect does not oppose any resistance to the passage of the wave at low frequencies. An interesting phenomenon appears for speciﬁc defect mass. The studied transmission probabilities present then strong asymmetric resonances. The origin of these can be attributed to the presence of resonant states induced by the defect whose frequency depends on the value of the adatom mass. Defects with m 0 > m gives rise to localized states below the optical modes; whereas defects with m 0 < m lead to local modes with frequencies larger the maximum frequencies of the acoustical or the optical modes [21]. The presence of these localized states in the frequency range of propagating states leads to the additional structures in the transmission spectra. They can be identiﬁed as Fano-like resonances. The transmission spectrum of the optical mode t33 in Fig. 4b shows such resonant states (indicated by arrows) in the case of the heavy defect mass. We further note that the frequency peaks in the vicinity of X @ 1.5 (Fig. 4b) are very close; the separation between the two localized states becomes very small and vanishes for large masses. Moreover, the resonance minima do not necessarily reach zero value but this is simply a facet of the graphical resolution in frequency. The results of the conductance K(X) are shown on Fig. 5. In addition to the curves of conductance relating to each defect mass considered previously, we also represented that of the perfect network (curve in indents em quadrats). In this case, i.e., neither without mass defect nor of force constants, the entering wave is totally trans-

Fig. 5. Total transmission probability versus dimensionless frequency in a double chain containing single impurity. The indents em quadrats line shows the conductance of the perfect waveguide, full line (m 0 = 3m) and dotted line (m0 ¼ m3 ).

mitted in each propagating mode. The conductance of the system becomes then important where the modes overlap. For this reason, its value reaches two units in the frequency range lain between X = 0 and X @ 1.42 then three in that located between X @ 1.69 and X @ 1.73. Owing to the fact that the resistance opposed to the passage of the wave in the perturbed region increases for a heavy defect, the overall transmission probability amplitude becomes more aﬀected than that referring to the light mass. This inﬂuence results also in a higher number of resonances (The same as that indicated by arrows on Fig. 4 in the case of heavy defect) because certain localized states, suitable for the light defect, are localized at higher frequencies than those of the implied propagating modes. A simple argument in favour of thispexplanation is contained in the frequency defﬃﬃﬃﬃﬃﬃﬃﬃﬃ inition x ¼ k=m of insulated harmonic oscillator which stipulates that a low mass corresponds well to a high vibrating frequency. The origin of these asymmetrical resonances is attributed, as in the electronic case, to a coupling of continuum-discrete localized states induced by the defect. We thus conclude that resonances take place at low frequencies for heavy defects and conversely for the light ones. These ﬁndings are in agreement with those of Tekman and Bagwell [7], who used a two mode-mode approximation. 4.2. Two separated adatomic scatterers We now consider the case of two defects separated by perfect waveguide piece and located at the positions l and r. The distance between the defects is d = (l r)a. For reasons of simplicity, we present the results for two identical perturbating masses, symmetrically laid out relatively to y axis. Fig. 6 shows the evolution of transmission probability in terms of the dimensionless frequency in the antisymmet-

Fig. 6. Transmission probability in the antisymmetric channel for the double chain containing two identical adatomic defects (m0 ¼ m2 ) separated by a distance d. The broken line gives the results for single defect.

M.S. Rabia / Journal of Molecular Structure: THEOCHEM 777 (2006) 131–138

rical acoustic mode t11 for diﬀerent distances in the case of light defect (m 0 = 0.3m). The dotted curve represents the elastic transmission probability in the case of a single adatomic scatterer. For a distribution of such defects, additional Fabry-Pe´rot oscillations due to the interference between multiply scattered waves in perturbed region are expected. It can be seen in Fig. 6 that their number in a given frequency interval increases as the gap widens. This number depends closely on the full width d of the defect region which is always multiple of the network parameter a. The amplitude of the oscillations becomes more signiﬁcant in the zone boundary. Moreover, the transmission curve does not present any Fano resonance. This can be explained by the fact that the respective eigenfrequencies of local defects are all positioned beyond the propagating band of the symmetrical acoustic mode [21]. The whole energy is quasi transmitted at the low frequencies as mentioned previously. Our results show clearly that the two separated scatterers have all the characteristics of FabryPe´rot resonator. As may be expected, the transmission spectrum obtained for heavy defect mass (Fig. 7) gives two Fano resonances which are associated with the two localized states induced by the impurity while interacting with the continuum of propagation (the peaks of resonance are indicated by arrows on the ﬁgure). We further note that the frequencies interval between peaks of resonances is reduced with the distance d and vanishes obviously for large distances. This explains the inﬂuence of the distance on the relative position of the respective transmission minima. For adjacent perturbations (d = a), the transmission probability remains strongly aﬀected even in the limit of extremely low frequencies and approaches 1/2 rather than unity when X ﬁ 0. This phenomenon seems rather strange since the transmission of sound in

Fig. 7. Transmission probability in the antisymmetric channel for the double chain containing two identical adatomic defects (m 0 = 5m) separated by a distance d. The broken line gives the results for single defect.

137

the low-frequency regime is not hindered by local defect. It can, however, be explained if we reconsider the functional behaviour of antisymmetric modes (Fig. 3) in the low-frequency range. In the limit X ﬁ 0, there are two modes, one propagating and the other evanescent but with Z ﬁ 1. The evanescent mode becomes more and more extended and at X = 0, it is degenerate with the acoustical propagating mode. For large distances (see, e.g., the case d = 10a), separation between the two states becomes very small. We further note that the peak of resonance could be overlapped with the Fabry-Pe´rot oscillation structure. Moreover, the maximum of resonance does not reach necessarily the unit value, as is usually the case. The number of Fabry-Pe´rot oscillations is also lower than the network parameters number a contained in full width d of the defect region but it is simply related to a resolution problem in the used frequency. Note that on average, the Fabry-Pe´rot oscillations follow a curve whose global shape is similar to that of the isolated defect in both cases. The transmission curves in the antisymmetric acoustical mode for three sequences of regularly spaced adatomic defects are shown on Figs. 8a to c. The mass of defect is

Fig. 8. Transmission probability in the antisymmetric channel for the double chain containing three defect sequences labelled (a), (b) and (c) in the case of light mass (m0 ¼ m3 ). The atomic structure scheme is given above each ﬁgure. The dashed curve is associated to two defects with the same spacing d = 3a.

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M.S. Rabia / Journal of Molecular Structure: THEOCHEM 777 (2006) 131–138

m 0 = 0.5m. The atomic structure schemes are given above each ﬁgure. Hear the Fabry-Pe´rot oscillations lead to a splitting into separated band of high transmission probabilities which are closely correlated with the distance between adjacent defects (d = 3a in this case), whereas the small oscillations within these windows depend directly on the full width of the defect region. If p indicates the number of isolated adatomic defects contained in the sequence, the structures appearing in each transmission spectrum are subdivided into (p 1) substructures (indicated by arrows in the ﬁgure). The substructures number is always equal to the number of network parameters contained in the full defect region width. Same results are observed by V. Pouthier et al. [10] on the transmittance spectrum of a nanowire containing a set of linear clusters separated by diﬀerent spacing. It is clear that the produced bands occurring for each defect type result from the spatial distribution of these defects in the perturbed zone. As the number of defects increases, we obtain again a new periodic structure characterized by band gaps. The bands could be considered as the signatures of these gaps. To understand this feature one would need, however, to determine the dispersion curves for a periodic distribution of defects. Some rapid oscillations in the frequency range boundary are due to the resonances caused by the defect-induced states attached to the symmetric optical mode. It should be noted that the global shape of these transmission curves is quite similar to that obtained in the case of two separated defects by a distance d = 3a (in dotted line on the ﬁgure). 5. Conclusion Following the approach of Landauer, we have recourse to the matching method to treat the scattering of elastic waves by perturbed multichannel quantum wires. In this way, we solve directly the Newton dynamical equation within the harmonic approximation framework in applying the scattering boundary conditions. The undulatory behaviour of the wave through defect was analyzed by taking account of defect mass variation and the bonding strengths in the perturbed region. We have analysed phonon scattering by isolated defect (single adatom), two separated adatoms and adatoms sequences. Numerical results show that the presence of defect in a double quantum wire modiﬁes particularly its mechanical and vibrational properties by the creation of new localized states and by a bulk phonon scattering [12] and surface phonon scattering [11]. Its inﬂuence results in a general decrease of the transmission probability amplitude accentuated by Fabry-Pe´rot oscillations (due to the interference between multiply reﬂected waves in the perturbed region) and Fano-like resonances (coherent coupling between propagating transmitted modes and localized-induced-defect states). As may be expected, the inﬂuence of the defects is smallest

in the acoustical regime and backscattering becomes most signiﬁcant near the zone boundaries. The interference eﬀects are of interest for improvements in the design of transducers and noise control [22] whereas Fano-type resonances are commonly used to build ﬁlters [23]. While measurements of mode-speciﬁc transmission probabilities will be rather diﬃcult, direct measurements of the total transmission (or conductance) spectrum should be feasible. The experimental challenge would be to couple a receiver and an emitter with known frequency characteristics to the ends of a waveguide avoiding back reﬂections at the junctions. The spectra of conductance can thus be regarded as ﬁngerprints of the speciﬁc defect structure and therefore be used for the characterization. In spite of their diﬀerent character, the scattering of vibrational waves and the scattering of electron waves can be described in terms of basically the same interference mechanism, namely Fabry-Pe´rot oscillations and Fano-like resonances. References [1] H. Ibach, D.L. Mills, Electron Energy Less Spectroscopy and Surface Vibrations, Academic, New York, 1982. [2] B. Kramer, Quantum Coherence in Mesoscopic Systems, plenum, New York, 1991. [3] R. Landauer, Z. Phys. B68 (217) (1987) 8099; J. Phys. Condens. Matter 1 (1989) 8099. [4] M. Bu¨ttiker, Phys. Rev. Lett. 57 (1986) 1761. [5] V.R. Velasco, F. Garcia-Moliner, L. Miglio, L. Colombo, Phys. Rev. B38 (1988) 3172–3179. [6] J. Szeftel, A. Khater, J. Phys. C 20 (1987) 4725. [7] E. Tekman, P.F. Bagwell, Phys. Rev. B48 (18) (1994) 299. [8] A. Virlouvet, H. Grimech, A. Khater, Y. Pennec, K. Maschke, J. Phys. Condens. Matter 8 (1996) 7589. [9] M. Belhadi, O. Raﬁl, R. Tigrine, A. Khater, J. Hardy, A. Virlouvet, K. Maschke, Eur. Phys. J. B15 (2000) 435–443. [10] V. Pouthier, C. Girardet, Phys. Rev. B66 (2002) 115322. [11] M.S. Rabia, H. Aouchiche, O. Lamrous, Eur. Phys. J. – A. P. 23 (2003) 95–102. [12] A. Fellay, F. Gagel, K. Maschke, A. Virlouvet, A. Khater, Phys. Rev. B55 (1997) 1707. [13] V. Pouthier, C. Girardet, J. Chem. Phys. 112 (2000) 5100–5104. [14] F. Gagel, K. Maschke, Phys. Rev. B52 (1995) 2013. [15] E. Mele, M.V. Pykhtin, Phys. Rev. Lett. 75 (1995) 3878. [16] T.E. Feuchtwang, Pys. Rev. 155 (1967) 731. [17] A.A. Maradudin, E.W. Montroll, G.H. Weiss, I. Patova, Theory of lattice Dynamics in the Harmonic Approximation, Academic Press, New York and London, 1971. [18] A. Khater, N. Auby, D. Kechrakos, J. Phys. Condens. Matter 4 (1992) 3743. [19] L. Landau, E. Lifshitz, The´orie de l’Elasticite´, Mir, Moscow, 1967, p. 146. [20] Y. Pennec, A. Khater, Surf. Sci. Lett. 348 (1996) L82. [21] A. Kumar, P.F. Bagwell, Phys. Rev. B43 (1991) 9012. [22] M.S. Kushwaha, A. Akjouj, B. Djafari-Rouhani, L. Dobrzynski, J.O. Vasseur, Solid State Commun. 106 (1998) 659. [23] M. Guglielmi, F. Montauti, L. Pellegrini, P. Arcioni, IEEE Trans. Microw. Theory Technol. 43 (1995) 1991.

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