Transactions Template JOURNAL OF ENGINEERING RESEARCH AND TECHNOLOGY, VOLUME 2, ISSUE 4, DECEMBER 2015 203 Coherent Phonons Scattering by Interstitial Impurities in Quasi- Planar crystals Mohammed Said Rabia Mouloud Mammeri University, BP 17 RP Hasnaoua II, Tizi-Ouzou 15000, Algeria, m2msr@yahoo.fr Abstract—using the matching method formalism, this work presents the transmission and reflection coefficients of coherent phonons in a quasi 2D quantum waveguide perturbed by reticular defects as interstitial impurities. Our waveguide is modelled by two infinite atomic chains. The implied interactions refer only to the bonding strengths between nearest and next nearest close neighbours. .Numerical results show that the transmission spectra exhibit Fano-like resonance features which result from degeneracy of localized-impurity states and propagating continuum modes. In addition, the scattering by multiple impurities induces interferences between diffused and reflected waves in the defect region giving birth to Fabry-Pérot oscillations. This interference phenomenon could provide an interesting alternative to investigate structural properties of materials. The results could be also useful for the design of phonon devices. Index Terms—Mesoscopic Disordered Systems; Reticular Dynamics; Phonons Scattering; Defect in Nanostructures, Matching Method; Numerical Simulation. I INTRODUCTION The survey of scattering and localization phenomena in the disordered mesoscopic systems interested the researchers at all times [1-3] because of the numerous applications found in classic metallurgy, in electrochemistry, in catalysis and in electronics. Our present knowledge of the related phenomena has been given by the work of Landauer [4], in which the studied sample is represented by a set of scatterers (reticular defects) inserted in bulk or on surface of crystalline structure. He showed that the conductance of a quantum wire is bound directly to the scattering properties of such system, consid- ered as a waveguide perturbed by defects. His approach has stimulated many researchers [5-10] to look for the effects of quantum coherence, most of the time by numerical methods, in dc transport particularly. Actually these phenomena are of renewed interest owing to advances in nanotechnologies, the basic motivation being to understand the limitations that reticular disorder may have on mechanical and vibrational properties of crystalline materials. In the present work, we study the phonons scattering by an interstitial impurity localized in an infinite double atomic chain. We analyze the behaviour of a plan wave which prop- agates throughout this crystal which is assimilated to a qua- si-planar crystallographic waveguide. We concentrate in calculating the reflected and transmitted parts of the inci- dental wave, the phononic conductance as well as the dis- placements of the irreducible atoms composing the per- turbed region. We are also interested by the determination of the localized induced impurity states especially important for transmission spectra interpretation. Different defect con- figurations are considered. The mathematical treatment of the problem resorts to the matching method [7,11] in the harmonic approximation framework [12-14] while using scattering boundary conditions. II STRUCTURAL MODEL The considered model consists of two linear parallel pe- riodic chains of masses, assimilated to a quasi-one- dimensional planar waveguide in which interstitial impuri- ties are incorporated. The parallel chains are composed of specific masses aligned along the direction of propagation (x axis). The situation is depicted in Figure 1. Each mass is linked to its nearest and next nearest neighbours by harmon- ic springs of stiffness constants k 1 and k 2 . The additional constants as K lv , k l and k v , are represented on the figure. To simplify, the distances between adjacent masses are consid- ered equal in the two Cartesian directions x and y of the plan. Also, to take account of the modification of the bond- ing strength field in the perturbed region (grey area M), we introduce a proportionality factor λ which indicates the ratio of the different force constants between the defect zone masses and those of the perfect lattice areas G (left) and D (right-hand side) located in sites separated by equivalent distances. III MATCHING METHOD PRINCIPLE Initiated by Feuchtwang in the sixties then revisited by Szeftel and al. in the eighties, the matching method returns account in a satisfactory way for the phonons dispersion curves [7-9] and for surface resonances. It gives also a more mailto:m2msr@yahoo.fr Mohammed Said Rabia / Phonons Scattering by Interstitial Impurities (2015) 204 general definition of the resonance concept and allows a more transparent analysis of the displacements behaviour in the vicinity of the Van Hove singularities [15]. However, its execution requires the crystal subdivision in three distinct regions having all the same periodicity along the surface. The procedure was described in details in references [8]. We will just present the necessary stages to the comprehension of the results analysis. A Perfect lattice dynamics For an atom occupying the site (l) and vibrating at the frequency  , the equations of motion can be written, using the harmonic approximation framework [14], in the follow- ing form:               ll lulu d rr llk lulm ' 22 2 22 ,',', ,       (1) where  and  represent the yx, directions of the plan;   mlm  indicates the atom mass located at site l ; r is the component of the relative position vector between sites l and 'l , d the distance separating them and  ',llk the bonding strength constant between the two atomic sites. Taking into account the problem symmetry and applying the scattering boundary conditions for which we get plan wave solutions, the perfect lattice atom equation of motion (1) rewrites itself in following matrix system:    0, 22   urZDI (2) where 1 22 km is the dimensionless frequency, I the identity matrix, )2,( rZD the )33(  dynamical matrix of the perfect lattice and u the vector displacement. The 2r parameter denotes the force constants ratio between nearest and next-nearest neighbours. The scattering problem in presence of defects imposes the knowledge of both propagating modes ( 1Z ) and eva- nescent ones ( 1Z ) of the perfect waveguide. In other words, for a given frequency, all solutions are necessary even those whose module is lower than unity. These solu- tions can be obtained by increasing the eigenvectors basis: 8,,5;)( 1 )(     lu Z lv . (3) We then rewrite equation (4) in the Z eigenvalue problem form,          )( )( with)( lv lu WWBZWA     , (4) where A and B are (4 4) matrices coming from the basis change. Let us note that the dimension of this generalized eigenvalue problem is twice as large as the original problem. B Coherent phonons scattering at defects Since the perfect waveguides do not couple between different eigenmodes, we can treat the scattering problem for each vibratory eigenmode separately. Generalization to every combination of these modes does not pose a particular problem. For an incoming wave from the left of Figure 1 in the eigenmode  , form must accompany your final submission. Authors are responsible for obtaining any security clearances. 1;)(  iuZV ii in   , (5) where Z is the attenuation factor of the entering mode, u  its eigenvector; the superscript )1( i indicates the site occupied by the atom with respect to the direction of propagation. … -2 -1 0 N N+1 N+2 ... G D M Matching Régions Figure 1: Schematic representation of a planar quasi-1D waveguide made up of two linear infinite chains perturbed by interstitial defects. The grey area M indicates defect region, G and D two semi infinite perfect waveguides. kl1 kv k2 Klv kl k1 y x Incident wave Transmitted wave Reflected wave Mohammed Said Rabia / Phonons Scattering by Interstitial Impurities (2015) 205 The resulting scattered waves are composed of a reflected and transmitted parts, which can be expressed as a superpo- sition of the eigenmodes of the perfect waveguide at the same frequency, i.e., 1;)(.. 11       iuru Z i Z i r       , (6)   2;)(..   iZuZtu ii t     , (7) where r and t indicates the reflection and transmis- sion coefficients normalized beforehand by group velocities (slopes of the dispersion curves) of the plan 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 (6) and (7), we can rewrite the dynam- ical equations for the perturbed double chain. Since there are perfect waveguides in regions G and D, we only need to solve Eqs. (1) 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. (6) and (7). Isolating the inhomogeneous terms describing the inci- dental wave, we obtain an inhomogeneous system of linear equations      inff VrDXRrD  ),,(),,( 22   (8) where ),,( 2 rD f  indicates the dynamical defect matrix, X  the vector gathering all the problem unknowns, inV  the incidental vector and R the matching matrix. As example, for an isolated defect we obtain a dynamical matrix  2618fD ; from where a matching matrix  1826R is deduced. Then the vector X  will be com- posed of eighteen unknowns including the ten displacements )(lu of the irreducible atoms, the four transmission coef- ficients and four reflection ones. IV IRREDUCIBLE ATOMS VIBRATION SPECTRA In combining the matching procedure to the Green’s functions and for a given wave vector parallel to the direc- tion of propagation [16], the matrix phonon spectral density reads ,)(2)( 22')',( ),(   m m l i l i ll PP   (9) where (l) ands (l’) are two atomic sites, α and β designate two different Cartesian directions and l iP is the component in direction α of the polarization vector of the atom (l) for the mode having a frequency Ωm. The vibration density of states (DOS) per atomic site Ni(Ω) in the perturbed defect region could be calculated by summing over the trace of the spectral density matrix (G (Ω 2 +jε) = ((Ω 2 +jε)I-Df (r2,λ)) -1 being the Green operator), i. e for (l)= (l’),                      )(Imlim 2 )( 2 0 jGN ll l (10) We strongly encourage authors to carefully review the material posted here to avoid problems with incorrect files or poorly formatted graphics. V RESULTS AND DISCUSSION Phonons scattered by impurity are analyzed relatively to an incidental wave coming from the left of Figure 1, with unit amplitude and a zero phase on the border atom (-1) located just at the beginning of the defect region M. A Single impurity scatterer The numerical results for the transmission and reflection coefficients in terms of the incident phonon frequency are consigned in Figure 2 in the case of an impurity mass mm 5.1' . We notice that the presence of the interstitial defect leads to a general decrease of the probability amplitude. As ex- pected, the influence of the impurity is relatively small in the acoustical regime because of the low implied frequencies. For 0 , we get 1T ; the subscript  (=1 to 4) refers to the number of eigenmodes characterizing the double at- tomic chain [7-9]. Moreover, the transmission spectra are marked by pronounced typical Fano-like resonances (null transmission in Figures 2). These asymmetric resonances can be attributed to the presence of impurity-induced reso- nant states, whose frequency depends on the value of the bonding forces in the defect region M. Consequently, these resonances take place at low frequencies for heavy defects and inversely for the light ones. These findings are in good agreement with those of Tekman and Bagwell [2], who used a two mode-mode approximation. Frequency Ω Figure 2: Transmission (full line) and reflection (doted line) coefficients vs. the phonon frequency for an isolated interstitial impurity mass m’= 1.5m. The dashed curve shows the good complementarity between the two coefficients. T1 R1 T2 R2 T3 R3 T4 R4 Mohammed Said Rabia / Phonons Scattering by Interstitial Impurities (2015) 206 Lastly the well known theoretical relation translating the conservation of energy principle,   1  TR (11) is fortunately satisfied and always checked for each frequen- cy (dashed lines in Figures 2). Besides, this condition consti- tutes an effective control method of the results. The results of the conductance )( are shown on Fig- ure 3. In addition to the curves of conductance relating to each impurity mass considered, we also represented that of the perfect lattice (dash-dotted histogram). In this case, the entering wave is totally transmitted in each propagating mode. The conductance of the system becomes then more important where the modes overlap. For this reason, its val- ue reaches more than unity in the concerned frequencies ranges. Otherwise the conductance spectrum is much more affected in the case of light impurity mass (dashed curve). In addition to resonances, this influence is translated by a less amplitude compared to for bigger masses at weak frequen- cies. B Extended defect The increase of the defect region width doesn't bring any- thing of qualitatively new in relation to the case of the single impurity. The addition of impurities results solely in the increase of the size of the linear system (8), but the matrix D ~ keeps its structure. The supplementary blocks have the same shape as those characterizing a single defect. We have limited our study to only ten interstitial impurities which already generates a ( 8072 ) defect matrix dimension. The effects described previously in the case of isolated step ap- pear, but they are even more difficult to isolate because of the biggest number of peak-dip structures near in frequen- cies. It is why we are not going to study in details these regions. On the other hand, we will limit ourselves to pre- sent a more global change of the transmission curves, pro- voked by the Fabry-Pérot oscillations issued from interfer- ences between the multiple scatterings of propagating states in the perturbed region. The phonon scatterings, considered for an extended defect composed of several interstitial impurities, are presented in Figure 4 for heavy impurity mass (m’=1.5m) in the anti- symmetric mode 1 [8]. It can be seen that the transmission curves structure became richer of several peaks. We observe a drastic dependence of Fabry-Pérot oscillations with the number of impurities. However, the number of main dips remained the same corresponding to the total number of lattice parameter a contained in the width of the perturbed region. The fact that their number seems to be lower on the figures is simply related to a resolution problem in the implied fre- quency range. Same results are observed by V. Pouthier and al. [17] on the transmittance spectrum of a nanowire con- taining a set of linear clusters separated by different spacing. Otherwise, the upper level of the Fabry Pérot oscillation can merge with the Fano-resonance peak. It should be noted that on average the global shape of the transmission curves is quite similar to that obtained in the case of an isolated impu- rity (in dashed line on the figures). The transmission curves are turned into a number of peak-dip structures, the reason is that the modes will inter- fere with each other due to the multiple reflections of the phonon waves in the perturbed region. In general, the multi- ple interferences in the perturbed waveguide imply the more complex transmission spectra. These interferences between multiply scattered waves result in Fabry–Pérot oscillations of increasing amplitudes with the frequency and whose number depends intimately of the number of impurities. Similar results are obtained in the study of adatomic defects [8,9,17-20] and substitutional defect columns [8] in the perturbed double quantum chain. Defects are separated by Frequency Ω Figure 3: The total transmission probability vs phonon frequency for impu- rity masses m’=0.5m (dashed line), m’=m (dotted line) and m’=1.5m (full line) in the case of a single impurity scatterer. The dashed-dotted histogram represents the total hypothetical phonon transmission capacity of the sys- tem. C o n d u c ta n c e Λ (Ω ) Figure 4: Transmission coefficient as a function of the phonon frequency for an extended defect composed of N defects of impurity mass m’=1.5m. The dashed curve refers to an isolated scatterer having the same mass. Frequency Ω T ra n sm is si o n p ro b a b il it y T 1 Mohammed Said Rabia / Phonons Scattering by Interstitial Impurities (2015) 207 different spacing in both configurations. C Phonons densities of states In Figure 5 are shown the phonon densities of states (DOS) versus the non-dimensional frequency for defect irreducible atoms (see Fig. 1). The results were calculated, according to Eq. (10), for two inhomogeneity masses light (m’=0.5m) in dashed line and heavy (m’=2m) in full line in the case of stiffened force constants. Due to obvious symmetry effects, analogous behaviors are observed for both columns (0) and (1). Therefore, the density of states is shown only for atom (0,0). It can be seen that spectra for this kind of lattice atoms is quite similar and present five main features as in the over- all transmission spectrum, mainly at resonant frequencies Ω≈1, 1.4, 1.7, 2.0 and 2.3 for heavy impurity mass. For light mass, the resonant peaks happened at frequencies Ω≈1.0, 1.4, 1.7 and 2.4. These resonances can be attributed to the presence of defect-induced resonant states. The phonon modes of the impurity atom reveal four signifi- cant resonant peaks (two for light impurity mass) at high frequency. These strongly localized modes are due to inter- stitial defect induced states. These peaks correspond mainly to the longitudinal modes near the high Brillouin zone boundary. The low-frequency peaks are mainly contributed by the transverse modes. As previously, these resonant peaks shift to higher (lower) frequencies for smaller (larger) impurity mass as expected. V CONCLUSION In this work, we have analyzed the behaviour of elastic waves propagating through a quantum waveguide perturbed by inter- stitial impurities. Our calculation resorts to the matching proce- dure based on the Landauer-Büttiker approach. The scattering is considered for isolated and extended impurity defects. In both configurations, strong asymmetrical resonances are observed in the transmission spectra; these structures, identified to Fano resonances, describe usually the interference between a propagating transmitted mode and a local defect mode. The resonance frequency depends closely of the impurity mass (or of the bonding force constants) in accordance with the relation Ω 2 =(k/m) defining the harmonic oscillator frequen- cy. For extended defects, resonance peaks and their number are determined by the perturbed region width given by the number of impurities. Moreover, the transmission spectrum is also characterized by other oscillations of Fabry-Pérot type due to the interferences between transmitted ant reflected waves in the perturbed region. Also their number depends closely of the defect region width. The transmission spectra can thus be used for identifying defects of specific structures and then being used for their char- acterization. The interference effects are of interest for im- provements in the design of transducers and noise control [21] whereas Fano-type resonances are commonly used to build filters [22]. The results could be also useful for the design of phonon devices. ACKNOWLEDGMENT I wish to thank Pr. S. Bouarab and Pr. M. Bennaki for their encouragements. REFERENCES [1] B. Kramer, Quantum Coherence in Mesoscopic Systems, 1991 (plenum, New York). [2] E. Tekman and P. F. Bagwell, Fano resonances in quasi-one dimensional electron waveguide, Phys. Rev., B48, 18 299, 1993. [3] Yu.A.Kosevich, Vibrations Localized near Surfaces and Interfaces in Nontraditional Crystals, Prog. Surf. Sci. vol 55, 1, 1997. [4] R. Landauer, Electrical Transport in Open and Closed Systems, Z. Phys. B 68, 217, 8099, 1987; Conductance determined by transmission: probes and quantised constriction resistance, J. Phys. Condens. Matter, 1, 8099, 1989. [5] M. Büttiker, Four-terminal phase-coherent conductance, Phys. Rev. Lett., 57, 1761, 1986. [6] E.S. Syrkin, P.A. Minaev, A. G. Shkorbatov and A. Feher, Influence of Interfacial layers on resonance phonon transport, Microelec. Eng. vol. 81 p 503, 2005. [7] J. Szeftel and A. Khater, Calculation of surface phonons and resonances: the matching method revisited I, J. Phys C, 20,4725 (1987) [8] A. Fellay, F. Gagel, K. Maschke, A. Virlouvet and A. Khater, Scattering of vibrational waves in perturbed quasi-one- dimensional multichannel waveguides, Phys.Rev, B55, 1707, 1997. [9] M. S. Rabia, Surface defect characterization in quantum wires by acoustical phonons scattering, J. Mol. Struc-Theochem, 777, 131-138, 2006. [10] A. Khater, N. Auby and D. Kechrakos, Surface-surface pho- non scattering by surface inhomogeneities, J. Phys. Condens. Matter, 4, 3743-3752, 1992. a) Atom (0,0) Frequency Ω Figure 5: Density of states (DOS) of the irreducible atoms versus incident phonon frequency for a) light (m’= 0.5 m, dotted line), and b) heavy (m’= 2 m, full line) impurity masses. D e n si ty o f S ta te s (a rb . u n it s) b) Impurity Mohammed Said Rabia / Phonons Scattering by Interstitial Impurities (2015) 208 [11] T. E. Feuchtwang, Dynamics of a semi-infinite crystal in a quasiharmonic approximation. II. the normal mode analysis of a semi-infinite lattice, Phys. Rev., 155, 731, 1967. [12] R. F. Wallis, Surface effects on lattice vibrations, Surf. Sci. 2, 146, 1964. [13] A. A. Maradudin, and J. Melangailis, Some dynamical proper- ties of surface atoms, Phys. Rev. 133, A1188, 1964. [14] A. A. Maradudin, E. W. Montroll, G. H. Weiss and Ipatova, Theory of lattice Dynamics in theHarmonic Approximation, 1971 (Academic Press New York and London). [15] L. Van Hove, Phys. Rev. 89, 1189, 1953. [16] S.M. Grannan, S.E. Labov, A.E. Lange, B. Sadoulet, B.A. Young, J. Emes, in: M. Meissner, R.O. Pohl (Eds.), Phonon Scattering in Condensed Matter VII, 1993 (Springer-Verlag). [17] M. S. Rabia, H. Aouchiche and O. Lamrous, Elastic waves scattering by extended defect surmounting a perfect lattice plan, Eur. Phys. J. – A. P. 23, 95-102, 2003. [18] V. Pouthier and C. Girardet, Electronic transmission through a set of metallic clusters randomly attached to an absorbed nan- owire: Localization-delocalization transition, Phys. Rev. B 66, 115322, 2002. [19] W. Kress, F. W. De Wette (Eds), Surface Phonons, 1991 (Springer-Verlag, Berlin). [20] M. S. Rabia, Coherent phonons scattering by geometric defects in a planar quantum waveguide, J. Physica E 42, 1307-1318, 2010. [21] M. Guglielmi, F. Montauti, L. Pellegrini, and P. Arcioni, Cor- rection to ―Implementing transmission zeros in inductive- window bandpass filters‖, IEEE Trans. Microwave Theory Technol, 43, 1991, 1995. [22] M. S. Kushwaha, A. Akjouj, B. Djafari-Rouhani, L. Dobrzyn- ski, J.O. Vasseur, Acoustic spectral gaps and discete transmis- sion in slender tubes, Solid State Commun. 106, 659, 1998. Mohammed Said RABIA is graduated from Université du Québec à Montréal (UQAM), Montréal, Canada (B. Sc. A en technologie de la Mécanique) in 1980. He received the M.Sc. and Ph.D. de- grees from Mouloud Mammeri University of Tiz-Ouzou (UMMTO), Algeria in 2000 and 2008, respectively. He is currently an associate professor at the UMMTO and researcher at the LMSE Laboratory, Algeria. His research interests: elastic waves scattering in mesoscopic disordered systems, guidance and confinement of elastic waves in phononic structures, modelling and simulation. E- mail: m2msr@yahoo.fr