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Phonons transmission by thin films sandwiched between two similar fcc structures
Superlattices and Microstructures 85 (2015) 226–236
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Phonons transmission by thin films sandwiched between two similar fcc structures Ghania Belkacemi a, Boualem Bourahla a,b,⇑ a b
Laboratory of Physics and Quantum Chemistry, M. Mammeri University, BP17RP, 15000 Tizi Ouzou, Algeria Institute of Molecules and Materials du Mans UMR 6023, University of Maine, 72085 Le Mans, France
a r t i c l e
i n f o
Article history: Received 17 December 2014 Received in revised form 20 May 2015 Accepted 21 May 2015 Available online 22 May 2015 Keywords: Lattice dynamics Sandwich structures Transport properties
a b s t r a c t An analytical and numerical formalism are developed to study the influence of the sandwiched atomic films on the vibration properties and phonon transmission modes in fcc waveguides. The model system consists of two identical semi-infinite fcc leads joined by ultrathin atomic films in between. The matching technique is applied to calculate the local Green’s functions for the irreducible set of sites that constitute the inhomogeneous domain. Numerical results are presented for the reflection/transmission, total phonon transmittance and localized vibration states in considered fcc lattices. The results show that vibrational properties of the sandwich materials are strongly dependent on the scattering frequency, the thickness of the insured films, incidence angles and elastic boundary conditions. We note that some of the fluctuations, observed in the vibration spectra, are related to Fano resonances, they are due to the coherent coupling between travelling phonons and the localized vibration modes in the neighborhood of the nanojunction domains. The number of localized modes which interact with the propagating modes of the continuum is proportional to the number of the sandwiched Slabs in the interfacial zone. The results give also the effect of the sandwiched ultrathin films on elastic waves propagation by atomic interfaces in fcc lattices. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Over the past decades, computer simulations based on atomistic models have become a powerful tool for gaining fundamental knowledge of materials processes and properties, and for providing input data for continuum models and materials design [1,2]. But, the knowledge of the phonons localized at an interface can be useful, such as in the study of electrical transport through a semiconductor heterojunction, where the electron–phonon coupling plays an important role. This knowledge should also give a better understanding of the crystalline growth. The localized phonons provide non dispersive propagation phonon modes at the atomic interface. In real applications, heterogeneous material interfaces are inevitable and thermal properties, in particular, are susceptible to changes in lattice structure, elemental composition, and dimensionality. Today, the critical scale of many thermal interface problems is in the nanojunction zones and nanometer range [3–6]. This requires a modeling and interpretative framework that goes beyond Fourier heat conduction analysis, for at such scales, heat transfer is determined by the transport of quantum energy carriers-phonons and electrons. The use of such structures, in high technology, requires a necessary basic knowledge for the adequate control of their properties. Therefore it is of
⇑ Corresponding author at: Laboratory of Physics and Quantum Chemistry, M. Mammeri University, BP17RP, 15000 Tizi Ouzou, Algeria. E-mail address: [email protected] (B. Bourahla). http://dx.doi.org/10.1016/j.spmi.2015.05.024 0749-6036/Ó 2015 Elsevier Ltd. All rights reserved.
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interest to develop numerical calculations for appropriate models for these systems, notably for sandwiched thin films between atomic leads. The vibrational properties of mesoscopic systems of various natures and of well-controlled complex geometries are now available as result of the recent development of nanotechnologies [7]. In such systems, the elastic waves propagating via atomic interfaces or by atomic inhomogeneities show intriguing useful properties and they are for more recent interest [8,9]. The interfaces and the inhomogeneities in the atomic structure scatter the elastic waves of the unperturbed lattice, which can be a waveguide. The resonances in the scattering spectrum for a waveguide containing perturbed domains are a signature of the localization effects in the breaking symmetry zone and in the neighborhood of the atomic defects. There are a number of theoretical methods to deal with the effects of different type of atomic inhomogeneities on vibration properties. The matching method we use in our work has been extended with success to study the vibration states and transmission/reflection of elastic waves at isolated nanostructures in several disordered systems [10–15]. This technique allows us to deal with both aspects the localization of phonon modes and the scattering of elastic waves within the same mathematical framework. Here we present a model for the study of the vibration spectra and the influence of the sandwiched atomic ultrathin films in perfect cubic waveguides onto the propagating of elastic waves. The results are presented for central nearest neighbors interactions in the harmonic approximation. In the first case, the analysis is carried out for a single ultrathin film sandwiched between two fcc reservoirs. In the second case, we consider that the two fcc leads are joined by three ultrathin atomic films. Each ultrathin atomic film is made from five atomic layers. The objective is to give a basic understanding for the relation between the coherent phonon transmission via the atomic interfaces and the thickness of the sandwiched films. Our numerical results do not address any particular experiment. Nevertheless, it is interesting to note that the experimental systems at present which are most relevant for the theoretical study of phonon transport would probably be the break junctions and atomic point contacts. The present article is organized as follow. In Section 2, we present the two studied models A/B/A and A/B/A/B/A, and we describe the dynamics and phonon modes of a perfect fcc lattices. In Section 2.1, we begin by introducing the essential features of the applied formalism; we calculate the bulk phonon dispersion branches and the corresponding group velocities. In Section 2.2, we interest to the dynamical properties of the sandwich structures, we give a theoretical outline of the phonon scattering at the interface zones and nanojunction domains. In Section 3 some numerical results are given for the individual and the total phonons transmission and localized vibration states, for the two considered fcc sandwich lattices. The phonon transmission and the localized vibration states spectra are calculated as a function of phonon frequencies, incidence angles, and elastic boundary conditions. Our conclusions are also presented in this section.
2. Model description To formulate the problem of vibration properties and transmission/reflection phonon modes, and to illustrate the method, we consider the two sandwich structures schematized in Fig. 1. The systems consist of three parts: two leads A (two semi-infinite lattices) and the interfacial region.
z y
B
A x
A
d
(a)
A
B
A
B
A
d
(b) Fig. 1. Illustration of the general form of the sandwiched systems A/B/A and A/B/A/B/A used for the phonon transmission calculations in fcc lattices. The typical systems are obtained by joining together two identical semi-infinite fcc leads A, respectively, by: (a) Single atomic slab, (b) triple atomic slabs. Each slab is composed from five atomic layers.
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In a first system, we study the dynamical properties of two semi-infinite reservoirs (A) joined by a single ultrathin atomic film (B), composed by a finite number of atomic layers (system A/B/A). In order to understand the effect of the thickness of the sandwiched films in fcc structures, we have considered the second system; it corresponds to the situation where the two semi infinite fcc reservoirs are joined by a triple atomic ultrathin films, we obtain the structure A/B/A/B/A. In all cases, the ultrathin films are made from five atomic layers. We consider that the A and B lattices have the same cubic meshing. The elastic waves are scattered back from the interfacial regions and would be dissipated by the semi-infinite cubic reservoirs A. In the general geometry of the analyzed structure, shown in Fig. 1, the y and z axis are parallel to the interface boundary, whereas the x-axis is normal to the planes of the nanojunctions. The scattering region is defined by the interfaces between the slabs. In this zone, the interactions between atomic sites are characterized by the force constants different from those of the pure bulk. In the semi-infinite perfect lattices A and B, atoms of mass mA (mB) are connected to their nearest neighbors by uniform elastic force constants k(A A) and k(B B), respectively; whereas, in the interface domain, the interactions between lattices is labeled k(A B). In addition, we take k(A B) = k(B A). The constant k(A A) is defined to be the corresponding force constant far from the boundary, and also k(B B) and k(A B) as the force constants between the atoms on the atomic interface boundary itself. To calculate the normalized frequencies of the sandwiched cubic systems A/B/A and A/B/A/B/A, we define for the perfect leads and the scattering zone the following ratios:
m¼
mB ; mA
r1 ¼
k1 ðB BÞ ; k1 ðA AÞ
rd ¼
k1 ðA BÞ k1 ðA AÞ
ð1Þ
Since structural relaxation is known to induce an elastically homogeneous strain at the boundary, we assume that the parameters r1 and rd may describe the homogeneous modification of the elastic boundary. Many factors could affect the phonon transmission across the interfacial region in real materials, such as material differences across interfaces, atomic reconstruction, dislocations, defects, strain fields, elastic and inelastic phonon scatterings. In this work, we limit our consideration to a lattice-matched system with perfect atomic arrangements at interfaces. To calculate scattering properties of the two sandwiched systems, we use a method specifically developed to investigate electronic transport through phase-coherent structures [16–18]. Transport coefficients are calculated as a function of phonon frequency within the Landauer-Buttiker formalism [19] by employing an exact recursive matching technique. 2.1. Phonon dispersion in fcc waveguide Under the harmonic approximation [20], the linearized equations of motion of an atom located at site l are given by
x2 ml ua ðl; xÞ ¼
XX l–l
0
0
2
0
kðl; l Þðra rb =d Þ½ub ðl ; xÞ ub ðl; xÞ
ð2Þ
b
The indices a and b denote Cartesian co-ordinates, ml is the atom’s mass, and ua ðl; xÞ is the corresponding displacement 0 vector vibration, of the l atom. ra describes the a-component of radius vector, d is the distance between l and l0 , and kðl; l Þ is the elastic force constant between atomic sites l and l0 . Eq. (2) shall be applied systematically and successively to the perfect leads and the interfacial region. Suppose that l and l0 are inside the perfect waveguide A, and far from of the nanojunction boundaries, the dynamics of vibrational modes outside the interfacial region may be described by the travelling wave solutions of Eq. (2) [21]
½X2 I DðgA ; /y ; /z ÞjuA i ¼ j0i
ð3Þ 2
X is a dimensionless frequency given by X2 ¼ x , where x0 is a characteristic lattice frequency, x20 ¼ k1 ðAAÞ , and I is the cormA x2 0
responding unit matrix. DðgA ; /y ; /z Þ is the dynamic matrix characteristic of the perfect fcc waveguide lattice and gA is a generic phase factor between two neighboring sites along the x-direction, gA ¼ ei/x ¼ eiqx a . The two variables /y and /z are the normalized dimensionless wave-vectors, defined as /y = qya and /z = qza. For simplicity we take a as the lattice distance between all adjacent sites, as well as on the boundary, neglecting any relaxation effects in the considered waveguides. The triplet (qx, qy, qz) correspond to the reciprocal lattice wave-vector components. Consequently, the doublet (/y, /z) (equivalent to (qy, qz)) describe the incidence angles of the system. The solution of Eq. (3) leads to three eigenmodes m = 1, 2, 3 with corresponding eigenfrequencies Xm and eigenvectors ~ um , at given wave-vector components ð/y ; /z Þ, for the perfect fcc lattice. We can distinguish two different kinds of solutions, namely the propagating elastic modes for which jgm j ¼ 1, and the physically evanescent modes for which jgm j 1. The evanescent modes are of interest because whilst they do not transport any energy they are nevertheless necessary for the detailed understanding of the scattering effects at the nanojunctions. To illustrate the model calculation, we present in Fig. 2a the dispersion curves, Xð/x Þ, for the elastic waves propagating, as a function of the normalized wavevector /x (/x ¼ qx a), where qx runs over the first Brillouin zone of the perfect fcc system. In
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3
mode 1 mode 2 mode 3
2.5
2
1.5
1
0.5
0 -4
-3
-2
-1
0
1
2
3
4
x Fig. 2a. Typical phonon dispersion curves for the perfect fcc crystallographic waveguide, presented over the first Brillouin zone.
this figure, all the phonon modes are acoustical; they are characterized by the limiting branch X ? 0 when /x ? 0. As may be seen, the eigenmodes labeled i 2 {1, 2, 3}, from bottom to top in that order, are propagating modes in the following frequency intervals: X1 X2 = [0.00, X1,max = 2.00], X3 [0.00, X3,max = 2.80]. It may also be shown that the eigenmodes of the fcc perfect waveguide are either symmetric or asymmetric with respect to the x-direction elastic waves propagation. We note that two of the three modes are degenerate. 2.2. Sandwiched system dynamics To analyze the phonons scattering in the presence of the interfacial region, as that in Fig. 1a and b, we not only have to know the propagating modes, but we have also to consider the evanescent eigenmode solutions of Eq. (3) as well. In other words, to determine the localized vibration states and the coherent phonons transmission via the nanojunctions in the sandwich systems, we use all the solutions gi , including the jgi j ¼ 1 propagating modes and the evanescent modes with jgi j 1. Both the propagating and evanescent eigenmodes can be obtained using different procedures [22,23]. The exact solutions are obtained as a function of the dimensionless frequency X, the wavevector ð/y ; /z Þ and the elastic interactions of the perfect fcc system. For an incoming elastic waves along the x-axis, in the eigenmode i, at given wave-vector ð/y ; /z Þ and frequency X, the scattering at the nanojunction boundary yields coherent reflected and transmitted elastic waves fields in the two semi-infinite A reservoirs. Given an incident elastic wave mode i from the left reservoir onto the interfacial region, the resulting Cartesian components a of the displacement field ~ u0a ðn; m; sÞ, for each fcc site, may be expressed using the matching approach [10–15], as the sum of the contributions coming from the incident elastic wave and from the fields backscattered at the nanojunction boundary, at the same frequency and wave-vector components
~ u0a ðn; m; sÞ ¼ gni~ ui þ
X
~ gn for n 0 j Rij uj ;
ð4Þ
j
ui and ~ uj denote the associated eigenvectors of the dynamic matrix for the perfect fcc system. The condition The vectors ~ n 6 0 refers the atomic layers before the interface domain. The notation Rij denotes the reflection coefficient which describe the scattering processes from the incident phonon mode i into the ensemble of the available eigenmodes j = 1, 2, 3 of the semi-infinite fcc perfect waveguide before the nanojunction domain. For sites inside the right hand reservoir A (semi-infinite lattice after the interfacial region), toward which the incident elastic wave mode i is transmitted, the resulting Cartesian components a of the displacement field ~ ua ðn; m; sÞ may be expressed by another appropriate superposition of the contributions from the forward scattered fields at the same the frequency and wave-vector components
~ ua ðn; m; sÞ ¼
X
gnj T ij~ uj ; for n d
ð5Þ
j
The condition n 6 d refers the atomic layers after the interface domain; d is the thickness of the interfacial region containing the sandwiched slabs. The notation Tij denotes the transmission coefficient which describe the scattering processes from the incident phonon mode i into the ensemble of the available eigenmodes j = 1, 2, 3 of the semi-infinite fcc perfect waveguide after the nanojunction domain. Let the vector j~ uint i regroup the atom displacements for the irreducible set of atomic sites on the interfacial region, where n 2 [0, d]. The atom dynamics equations for these sites coupled to those for neighboring sites in the matching domains of the
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reservoirs generate a rectangular system of equations with a number of unknown variables greater than the number of rows of this system. E Ei h T for the reflection and transmission proConsider next a Hilbert space constructed from the basis vectors j~ R ; ~ uint i; ~ uint i. The equations of motion coupled to the two reservoirs on both sides of the interface domain, may be cesses and from j~ E Ei h T . uint i; ~ R ; ~ written in terms of vector j~ Using the transformations connecting the displacement fields in Eqs. (4), (5), we obtain a square linear inhomogeneous system of the equations of the form
ih
h
E Ei
T uint i; ~ X2 I DðkAB ; /y ; /z Þ j~ R ; ~
! E ¼ IH ; gj ; /y ; /z
ð6Þ
! E Here the vector IH ; gj ; /y ; /z , mapped appropriately onto the basis vectors in the constructed Hilbert space and regroups the inhomogeneous terms describing the incoming elastic wave. DðkAB ; /y ; /z Þ is the matrix dynamic of the sandwich structure. Here kAB describes the elastic force interaction between sites of the two different lattices A and B. In the present analysis, by applying appropriate transformation, we obtain the matched atom dynamics matrix in the form
DðkAB ; /y ; /z Þ2121 ¼ M d ð21 27Þ MR ð27 21Þ for the A=B=A system and DðkAB ; /y ; /z Þ5151 ¼ M d ð51 57Þ MR ð57 51Þ for the A=B=A=B=A system; where Md and MR denote, respectively, the rectangular system of equations and the matching matrix. The solutions of Eq. (6) give explicitly the reflection, rij , and transmission, tij , coefficients in the studied sandwich structures, and yield consequently a complete description for the scattering processes at the nanojunctions for any given incident elastic wave. These are given by
8 2 < r ij ðX; /y ; /z Þ ¼ vv gi Rij gi : t ðX; / ; / Þ ¼ v gj T 2 ij y z v gi ij
ð7Þ
vgi is the group velocity of the eigenmodes i and Rij and Tij are, respectively, the reflection and the transmission probabilities. The scattering cross sections are normalized with respect to the group velocities of the elastic waves to ensure the unitarity of the scattering processes. For an evanescent phonon mode the corresponding group velocity is equal to zero. Typical results for the group velocities of the three eigenmodes are presented in Fig. 2b. Some modes show regions of anomalous dispersion where the group velocity is negative: these regions require a careful examination of the notion of forward and backward scattering for the elastic waves. The group velocity is related to the variation of the real wavenumbers accordingly to the frequency. But for imaginary wavenumbers, a null velocity is imposed [24]. In all cases the materials should have fully real modes. To determine this velocity, several methods can be used [25,26]. ! In this work, the group velocities are calculated directly from the slopes of the phonon dispersion curves, ~ v g ¼ gradXð~qx Þ. A challenge is encountered with this method at the locations where the dispersion branches cross. At these locations, it is difficult to keep track of which branch corresponds to each mode. Here, this challenge is avoided by following the approach of Wang et al. [27], and deriving an analytical expression for the phonon group velocity in terms of the dynamical matrix. This expression allows one to easily keep track of the frequency, group velocity, and polarization vector that correspond to a particular phonon mode. We can further define total reflection and transmission cross sections for a given eigenmode i, at given wave-vectors ð/y ; /z Þ and scattering frequency X, by summing over all the contributions of the scattered eigenmodes j, such that
X 8 r ðX; /y ; /z Þ ¼ r ij ðX; /y ; /z Þ > > < i j X > t ij ðX; /y ; /z Þ > : ti ðX; /y ; /z Þ ¼
ð8Þ
j
In order to describe the overall transmission and transmission cross sections of the sandwich systems at a given wave-vectors ð/y ; /z Þ and frequency X, it is useful to define the transmittance of the systems rðX; /y ; /z Þ, by summing over all input and output propagating phonon modes
rðX; /y ; /z Þ ¼
XX i
tij ðX; /y ; /z Þ
j
where the sum is carried out over all propagating phonon modes at frequency X and given wave-vectors ð/y ; /z Þ.
ð9Þ
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12
10 Vg 1 Vg 2 Vg 3
Vg
8
6
4
2
0
0
0.5
1
1.5
2
2.5
3
Fig. 2b. Curves of the group velocities of the different propagating modes (1–3) of the perfect fcc crystallographic waveguide Vg, as function of the scattering frequency X.
3. Numerical applications and discussion The scattering of the phonons at the interface between two similar fcc lattices is considered, in this work, with reference to elastic waves incident from the left reservoir to the right reservoir of the scattering region in Fig. 1. We examined two sandwich structures containing different number of atomic layers in the interfacial region. They are modeled respectively in Fig. 1a and b, where the thickness of the interfacial domain is defined by the distance d. For the two studied systems, the wavevector representation can be used in the transverse direction due to its infinite size along the interface domain [28]. The typical geometry of the first system A/B/A is presented in Fig. 1a. On the other hand, Fig. 1b gives the typical geometry of the second system A/B/A/B/A. In the two forms of the interface domain, we have simulated the case where each sandwiched slab is composed by five atomic layers. When the elastic wave arrives at the nanojunction zone, the fist one is split into its transmitted and reflected parts. Consequently, the nanojunction zones joining different types of atomic slabs are considered as effective atomic inhomogeneities in the studied structures. In this work, the model is applied to simulated the phonon scattering in the two considered fcc sandwich structures, by using Eqs. (6)–(9). The dynamic properties, the phonon transmission/reflection, localized vibration states and the total transmittance are calculated numerically only for comparable masses (mA mB). In addition, the analysis is carried out, also, for three different cases of the interface elastic constants kAB, where each case determines a choice of the elastic properties of the interface domain. These are defined, respectively, by: (i) kAB = 0.75, (ii) kAB = 1.00, (iii) kAB = 1.25. We remind that the interface elastic constant kAB is considered as the arithmetic mean of the A and B nearest neighbor interaction constants, so then kAB = (kAA + kBB)/2. The first case (i) corresponds to a softening of its elastic constants in comparison to the bulk of the lattice A; the second case (ii) corresponds to the situation where the elastic constants are the same, and in contrast the third case (iii) corresponds to a hardening of its elastic constants. These possibilities are similar to situations where changes in elastic constants may take place in the neighborhood of steps and kinks in surfaces [29]. The purpose of this numerical procedure is to investigate how the local dynamics respond to local changes in the elastic properties. To study the phonon transmission and reflection effects, the inhomogeneous Eq. (6) is solved numerically. The transmission spectra are calculated from Eq. (7), for a given incident angle ð/y ; /z Þ ¼ ð0; 0Þ. The results are presented in Fig. 3a for the system A/B/A and in Fig. 3b for the second system A/B/A/B/A. In Fig. 3, we observe that the transmission and reflection coefficients are presented as a function of the scattering frequency X and at a given direction for the three phonon modes (t1, r1), (t2, r2) and (t3, r3). The results are obtained for the three cases of elastic constants: rd = 0.75, 1.00 and 1.25. In all modes, we show that the transmission and the reflection coefficients verify the unitarity condition s = ti + ri = 1, (i = 1, 2, 3) of the scattering matrix and that this is used as a check on the numerical calculation. We note that all transmission coefficients are hopeless for X superior to 2.80. As expected the influence of the sandwiched films is nil at low frequencies, where X tends to zero, the transmission coefficients t1, t2 and t3, tend to her maximum values (1); in contrast, at the high frequency the coefficients are sensitive to the effect of the interface and become zero when X tends to 2.00 for the modes 1 and 2, and X tends to 2.80 for the mode 3. Another general characteristic of the phonon transmission is the displacement of its spectral features to higher frequencies with increasing hardening of the elastic force constant kAB between the A and B layers, for the two configurations, which is a signature of the nanojunction matching necessary for the transmission the elastic waves.
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Transmission and reflection coefficients
1.5
t3 r3 s3
1
1.5 t2 r2 s2
1
0.5
0.5
0 0
0.5
1
1.5
2
2.5
3
1.5
t1 r1 s1
1 0.5
0 0
0.5
1
1.5
2
2.5
3
1.5
0 0
1
1
1
0.5
0.5
0.5
0 0
0.5
1
1.5
2
2.5
3
0 0
0.5
1
1.5
2
2.5
3
0 0
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
0.5
1
1.5
2
2.5
1.5
0
0.5
1
1.5
2
2.5
3
0
3 t1 r1 s1
0.5
1
1.5
2
2.5
3 t1 r1 s1
0
0.5
1
1.5
2
2.5
3
Fig. 3a. Phonon transmission t1, t2, t3 (dotted lines) and reflection r1, r2, r3 (solid lines) coefficients and their sum s1, s2, s3 (dashed lines) across a single sandwich slab in the fcc structure A/B/A. They are given as function of the scattering frequency X, at a fixed vector components (/y, /z) = (0, 0).
1.5
Transmission and reflection coefficients
1.5
t3 r3 s3
1 0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
0
1.5
t2 r2 s2
1
0.5 0.5
1
1.5
2
2.5
3
0
0
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0
1.5
1.5
1.5
1
1
1
0.5
0.5
0.5
0
0
0
0.5
1
1.5
2
2.5
3
0
t1 r1 s1
1
0.5
1
1.5
2
2.5
3
0
0
0.5
1
1.5
2
2.5
3 t1 r1 s1
0.5
1
1.5
2
2.5
3 t1 r1 s1
0.5
1
1.5
2
2.5
3
Fig. 3b. Phonon transmission t1, t2, t3 (dotted lines) and reflection r1, r2, r3 (solid lines) coefficients and their sum s1, s2, s3 (dashed lines) across a triple sandwich slabs in the fcc structure A/B/A/B/A. They are given as function of the scattering frequency X, at a fixed vector components (/y, /z) = (0, 0).
Some of the spectral maxima in the interval X 2 [0, 2.80] correspond to characteristic Fano resonances, since the atom vibration states on the interfacial domain are effectively localized states embedded in the continuum of the waveguide incident phonon modes. In another words, this is attributed to the coupling of the propagating incident elastic waves with a localized vibration modes induced by the interface domain. However, we can conclude that the waveguide configured in Fig. 1b introduce more localized modes, i.e., more Fano resonances in transmission spectra than the structure configured in Fig. 1a. In addition, we observe that the interfacial phonon transmission decreases with increasing the thickness of the sandwiched films, which is well expected due to the multiple interface effects. We note that when phonons are transported in multilayered structures, the elastic waves can interfere with each other to form mini phonon bands and can thus change the phonon transmission. Moreover, similar transmission behavior has been observed for phonon radiative transport across periodic structures [30]. In Fig. 4, we plot the total phonon transmission rðX; /y ; /z Þ (or the phonon transmittance) across the interface domain, given by formula (9) for the direction ð/y ; /z Þ ¼ ð0; 0Þ. For the first sandwich system A/B/A, the phonon transmittance conductance is presented, in Fig. 4a, throughout the interval X 2 [0, 2.80], which corresponds to the interval frequency of propagation phonon modes from the first semi-infinite reservoir A to the second reservoir A after the insured thin film B. In the other hand, in Fig. 4b, we present the total phonon transmission via the interfacial domain of the system A/B/A/B/A, which are propagating also in the frequency interval X 2 [0, 2.80]. In the two cases of the sandwiched thin films examined in this work, the conductance spectra always starts with her higher values at low frequencies (X tend zero) and decrease with increasing X then becomes zero at the Brillouin Zone limit. The curves of the phononic transmittance present the pick resonances of different height and widths, what gives him a rough aspect.
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3.5
3
2.5
2
1.5
1
softening homogeneous hardening
0.5
0
0
0.5
1
1.5
2
2.5
3
Fig. 4a. The curves of the phonon transmittance for the fcc sandwich structure A/B/A, as function of the scattering frequency X, at the given direction (/y, /z) = (0, 0).
3.5
3
2.5
2
1.5
1 softening homogeneous hardening
0.5
0
0
0.5
1
1.5
2
2.5
3
Fig. 4b. The curves of the phonon transmittance for the fcc sandwich structure A/B/A/B/A, as function of the scattering frequency X, at the given direction (/y, /z) = (0, 0).
The phonon transmittance, rðX; /y ; /z Þ, is less than or equal to one phonon throughout the frequency interval propagation, for all two examined configurations. This illustrates how the sandwiched thin films at the heart of the nanojunction domain in each case constricts the transmission to a maximum of one phonon at a time, which is characteristic for the atomic layers with finite thickness or confined films n 2 [0, d]. In all cases, the conductance spectra always starts with her higher values at low frequencies and fluctuate for the intermediate values of X and decrease with increasing X then becomes zero at the Brillouin Zone limit. In Fig. 4a, the phonon transmittance via the atomic interface in the first system A/B/A is more important than the phonon transmittance in the second system A/B/A/B/A given in Fig. 4b; this can be explained by thickness of the sandwiched thin films and the velocity of decay of the corresponding wave-function which is different in the both sides of the interfaces domain. In each figure, we observe the systematic displacement of its spectral maxima to great values; this can be explained by the difference value of the elastic constant parameter kAB, and a situation that confirms the importance of the interface domain localized states in the transmittance spectra. Another interesting observation is the important magnon transmittance, rðX; /y ; /z Þ, in the interval X 2 [0, 2.00], for the two studied systems, and for all considered elastic constant kAB. The observed maxima in the interval mentioned interval frequency correspond to characteristic Fano resonances since the vibration states on the interface domain are effectively localized discrete states which interact with the continuum of the incident phonon waveguide modes. In particular, for the first system A/B/A, the features of the conductance spectrum attributed to Fano resonances, XF, correspond to the
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maxima that shift through the frequencies with increasing kAB. In each mode, the observed Fano resonance positions XF = 1.36, 1.74 (modes 1 and 2) and XF = 1.95, 2.45 (mode 3), for the softening elastic constant situation (case (i)) shift to the following higher frequencies XF = 1.44, 1.82 (modes 1 and 2) and XF = 2.05, 2.58 (mode 3) for the hardening situation, which corresponds to the case (iii). The quoted remarks are also valid for the second system A/B/A/B/A. The features of the Fano resonances are observed at the positions XF = 0.88, 1.38, 1.74 (modes 1 and 2) and XF = 1.26, 1.94, 2.45 (mode 3), for the case (i). With increasing kAB (case (iii)), XF = 0.94, 1.45, 1.82 (modes 1 and 2) and XF = 1.33, 2.05, 2.58 (mode 3). For the case (ii), which describes the situation where the elastic constants are same everywhere, a total phonon transmission are observed. In this case, the considered systems correspond to perfect lattices without inhomogeneities. We have also applied the model to calculate numerically the energies of the localized vibration states in the neighborhood of the interface zone between the two semi-infinite reservoirs A. These energies are obtained by resolution of the determinant det½DðkAB ; /y ; /z Þ ¼ 0 of the matrix mentioned in Eq. (6). They correspond to propagating phonons via the nanojunction zone, which are spatially localized in the sense that their atom vibration amplitudes decay along the x-direction inside the two semi-infinite reservoirs located on both sides of the interfacial domain. These localized vibration modes are induced by the breakdown of the translation symmetry engendered by the interface zone. Moreover, the localized modes couple with the incident propagating phonon modes in the scattering region. These localized states are presented in Figs. 5a and 5b in the form of bulk displacements for the perfect lattice and the dispersion curves of the irreducible sites of the interfacial region. These dispersion branches are given as a function /y, propagating parallel to the interface plane nanojunction. The purpose, of the calculations, of the elastic wave dispersion curves for A bulk and interface modes, is to demonstrate the essential features of the elastic wave interface modes and the influence of the parameter kAB on the localized modes of phonons at the interface zone. As has been pointed out in Section 2, where jgj ¼ 1 corresponds to the propagating modes in the semi-infinite A perfect regions as shown in the shaded area in Figs. 5a and 5b. Near the interfacial zone, the dispersion curves depict phonons propagating along the x-direction (normal to the interface) that are, however, effectively localized in the sense that their amplitude of the displacement is evanescent in the direction normal to the surfaces of the two semi-infinite reservoirs A, located on both sides of the interface domain. The amplitude of the localized elastic wave in the interface zone decays exponentially with increasing penetration into A subsystems. In Fig. 5b, the interface dispersion branches contains the dispersion branches of Fig. 5a, this can be explained by the fact that the effects of the sandwiched films are superimpozed. In addition, the localized states displace to higher energies with increasing elastic constant between the atomic of the two lattices A and B. We note that the number of the localized vibration states of the first system A/B/A is different from those of the second system A/B/A/B/A. This is explained by the increasing thickness of the sandwiched layers between the A reservoirs. Varying the parameters of the boundary domain can technically control the elastic waves transmission properties. This implies the possibility of a nanometric procedure to organize the heat transfer from a thermal or coherent source, preferentially, into different branches of a mesoscopic system.
3.5 softening case hardening case
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
y
Fig. 5a. Curves of the localized vibration sates on the boundary of the fcc sandwiched A/B/A system. The A bulk energies are shown shaded, whereas the circular symbols represent the interface modes.
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3.5 softening case hardening case
3
2.5
2
1.5
1
0.5
0
0
0.5
1
1.5
2
2.5
3
3.5
y
Fig. 5b. Curves of the localized vibration sates on the boundary of the fcc sandwiched A/B/A/B/A system. The A bulk energies are shown shaded, whereas the circular symbols represent the interface modes.
4. Conclusion In conclusion, we have studied the generic properties of phonon propagation and elastic waves diffusion by different atomic nanojunctions in cubic lattices by the matching technique in the harmonic approximation. Our results show that the phonon transmission across sandwiched ultrathin films in fcc lattices can be controlled by the thickness of the inserted films. Additionally, the phononic transmittance of the fcc sandwiched systems is a strong function of the scattering frequency, vector components and the atomic configuration. The formalism presented here is independent of the configuration of the interface region, which makes it easy to extend them to study a variety of disordered interface problems. At lower frequencies the individual and total transmission spectra start with its maximal values, present the fluctuation for 0 6 X 6 2.80 and become null at the Brillouin Zone limit. The fluctuations number related to Fano resonances are in good agreement with the localization theory. Our work demonstrates that the number of localized modes which interact with the propagating modes of the continuum is proportional to the number of the sandwiched films in the interfacial zone. This explains the higher number of fluctuations in the vibration spectra of the system with triple sandwich slabs. Acknowledgment The authors would like to thank Dr. Louiza Belkacemi from the University of Houston, TX, USA for editing our manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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Contents lists available at ScienceDirect
Superlattices and Microstructures journal homepage: www.elsevier.com/locate/superlattices
Phonons transmission by thin films sandwiched between two similar fcc structures Ghania Belkacemi a, Boualem Bourahla a,b,⇑ a b
Laboratory of Physics and Quantum Chemistry, M. Mammeri University, BP17RP, 15000 Tizi Ouzou, Algeria Institute of Molecules and Materials du Mans UMR 6023, University of Maine, 72085 Le Mans, France
a r t i c l e
i n f o
Article history: Received 17 December 2014 Received in revised form 20 May 2015 Accepted 21 May 2015 Available online 22 May 2015 Keywords: Lattice dynamics Sandwich structures Transport properties
a b s t r a c t An analytical and numerical formalism are developed to study the influence of the sandwiched atomic films on the vibration properties and phonon transmission modes in fcc waveguides. The model system consists of two identical semi-infinite fcc leads joined by ultrathin atomic films in between. The matching technique is applied to calculate the local Green’s functions for the irreducible set of sites that constitute the inhomogeneous domain. Numerical results are presented for the reflection/transmission, total phonon transmittance and localized vibration states in considered fcc lattices. The results show that vibrational properties of the sandwich materials are strongly dependent on the scattering frequency, the thickness of the insured films, incidence angles and elastic boundary conditions. We note that some of the fluctuations, observed in the vibration spectra, are related to Fano resonances, they are due to the coherent coupling between travelling phonons and the localized vibration modes in the neighborhood of the nanojunction domains. The number of localized modes which interact with the propagating modes of the continuum is proportional to the number of the sandwiched Slabs in the interfacial zone. The results give also the effect of the sandwiched ultrathin films on elastic waves propagation by atomic interfaces in fcc lattices. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Over the past decades, computer simulations based on atomistic models have become a powerful tool for gaining fundamental knowledge of materials processes and properties, and for providing input data for continuum models and materials design [1,2]. But, the knowledge of the phonons localized at an interface can be useful, such as in the study of electrical transport through a semiconductor heterojunction, where the electron–phonon coupling plays an important role. This knowledge should also give a better understanding of the crystalline growth. The localized phonons provide non dispersive propagation phonon modes at the atomic interface. In real applications, heterogeneous material interfaces are inevitable and thermal properties, in particular, are susceptible to changes in lattice structure, elemental composition, and dimensionality. Today, the critical scale of many thermal interface problems is in the nanojunction zones and nanometer range [3–6]. This requires a modeling and interpretative framework that goes beyond Fourier heat conduction analysis, for at such scales, heat transfer is determined by the transport of quantum energy carriers-phonons and electrons. The use of such structures, in high technology, requires a necessary basic knowledge for the adequate control of their properties. Therefore it is of
⇑ Corresponding author at: Laboratory of Physics and Quantum Chemistry, M. Mammeri University, BP17RP, 15000 Tizi Ouzou, Algeria. E-mail address: [email protected] (B. Bourahla). http://dx.doi.org/10.1016/j.spmi.2015.05.024 0749-6036/Ó 2015 Elsevier Ltd. All rights reserved.
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interest to develop numerical calculations for appropriate models for these systems, notably for sandwiched thin films between atomic leads. The vibrational properties of mesoscopic systems of various natures and of well-controlled complex geometries are now available as result of the recent development of nanotechnologies [7]. In such systems, the elastic waves propagating via atomic interfaces or by atomic inhomogeneities show intriguing useful properties and they are for more recent interest [8,9]. The interfaces and the inhomogeneities in the atomic structure scatter the elastic waves of the unperturbed lattice, which can be a waveguide. The resonances in the scattering spectrum for a waveguide containing perturbed domains are a signature of the localization effects in the breaking symmetry zone and in the neighborhood of the atomic defects. There are a number of theoretical methods to deal with the effects of different type of atomic inhomogeneities on vibration properties. The matching method we use in our work has been extended with success to study the vibration states and transmission/reflection of elastic waves at isolated nanostructures in several disordered systems [10–15]. This technique allows us to deal with both aspects the localization of phonon modes and the scattering of elastic waves within the same mathematical framework. Here we present a model for the study of the vibration spectra and the influence of the sandwiched atomic ultrathin films in perfect cubic waveguides onto the propagating of elastic waves. The results are presented for central nearest neighbors interactions in the harmonic approximation. In the first case, the analysis is carried out for a single ultrathin film sandwiched between two fcc reservoirs. In the second case, we consider that the two fcc leads are joined by three ultrathin atomic films. Each ultrathin atomic film is made from five atomic layers. The objective is to give a basic understanding for the relation between the coherent phonon transmission via the atomic interfaces and the thickness of the sandwiched films. Our numerical results do not address any particular experiment. Nevertheless, it is interesting to note that the experimental systems at present which are most relevant for the theoretical study of phonon transport would probably be the break junctions and atomic point contacts. The present article is organized as follow. In Section 2, we present the two studied models A/B/A and A/B/A/B/A, and we describe the dynamics and phonon modes of a perfect fcc lattices. In Section 2.1, we begin by introducing the essential features of the applied formalism; we calculate the bulk phonon dispersion branches and the corresponding group velocities. In Section 2.2, we interest to the dynamical properties of the sandwich structures, we give a theoretical outline of the phonon scattering at the interface zones and nanojunction domains. In Section 3 some numerical results are given for the individual and the total phonons transmission and localized vibration states, for the two considered fcc sandwich lattices. The phonon transmission and the localized vibration states spectra are calculated as a function of phonon frequencies, incidence angles, and elastic boundary conditions. Our conclusions are also presented in this section.
2. Model description To formulate the problem of vibration properties and transmission/reflection phonon modes, and to illustrate the method, we consider the two sandwich structures schematized in Fig. 1. The systems consist of three parts: two leads A (two semi-infinite lattices) and the interfacial region.
z y
B
A x
A
d
(a)
A
B
A
B
A
d
(b) Fig. 1. Illustration of the general form of the sandwiched systems A/B/A and A/B/A/B/A used for the phonon transmission calculations in fcc lattices. The typical systems are obtained by joining together two identical semi-infinite fcc leads A, respectively, by: (a) Single atomic slab, (b) triple atomic slabs. Each slab is composed from five atomic layers.
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In a first system, we study the dynamical properties of two semi-infinite reservoirs (A) joined by a single ultrathin atomic film (B), composed by a finite number of atomic layers (system A/B/A). In order to understand the effect of the thickness of the sandwiched films in fcc structures, we have considered the second system; it corresponds to the situation where the two semi infinite fcc reservoirs are joined by a triple atomic ultrathin films, we obtain the structure A/B/A/B/A. In all cases, the ultrathin films are made from five atomic layers. We consider that the A and B lattices have the same cubic meshing. The elastic waves are scattered back from the interfacial regions and would be dissipated by the semi-infinite cubic reservoirs A. In the general geometry of the analyzed structure, shown in Fig. 1, the y and z axis are parallel to the interface boundary, whereas the x-axis is normal to the planes of the nanojunctions. The scattering region is defined by the interfaces between the slabs. In this zone, the interactions between atomic sites are characterized by the force constants different from those of the pure bulk. In the semi-infinite perfect lattices A and B, atoms of mass mA (mB) are connected to their nearest neighbors by uniform elastic force constants k(A A) and k(B B), respectively; whereas, in the interface domain, the interactions between lattices is labeled k(A B). In addition, we take k(A B) = k(B A). The constant k(A A) is defined to be the corresponding force constant far from the boundary, and also k(B B) and k(A B) as the force constants between the atoms on the atomic interface boundary itself. To calculate the normalized frequencies of the sandwiched cubic systems A/B/A and A/B/A/B/A, we define for the perfect leads and the scattering zone the following ratios:
m¼
mB ; mA
r1 ¼
k1 ðB BÞ ; k1 ðA AÞ
rd ¼
k1 ðA BÞ k1 ðA AÞ
ð1Þ
Since structural relaxation is known to induce an elastically homogeneous strain at the boundary, we assume that the parameters r1 and rd may describe the homogeneous modification of the elastic boundary. Many factors could affect the phonon transmission across the interfacial region in real materials, such as material differences across interfaces, atomic reconstruction, dislocations, defects, strain fields, elastic and inelastic phonon scatterings. In this work, we limit our consideration to a lattice-matched system with perfect atomic arrangements at interfaces. To calculate scattering properties of the two sandwiched systems, we use a method specifically developed to investigate electronic transport through phase-coherent structures [16–18]. Transport coefficients are calculated as a function of phonon frequency within the Landauer-Buttiker formalism [19] by employing an exact recursive matching technique. 2.1. Phonon dispersion in fcc waveguide Under the harmonic approximation [20], the linearized equations of motion of an atom located at site l are given by
x2 ml ua ðl; xÞ ¼
XX l–l
0
0
2
0
kðl; l Þðra rb =d Þ½ub ðl ; xÞ ub ðl; xÞ
ð2Þ
b
The indices a and b denote Cartesian co-ordinates, ml is the atom’s mass, and ua ðl; xÞ is the corresponding displacement 0 vector vibration, of the l atom. ra describes the a-component of radius vector, d is the distance between l and l0 , and kðl; l Þ is the elastic force constant between atomic sites l and l0 . Eq. (2) shall be applied systematically and successively to the perfect leads and the interfacial region. Suppose that l and l0 are inside the perfect waveguide A, and far from of the nanojunction boundaries, the dynamics of vibrational modes outside the interfacial region may be described by the travelling wave solutions of Eq. (2) [21]
½X2 I DðgA ; /y ; /z ÞjuA i ¼ j0i
ð3Þ 2
X is a dimensionless frequency given by X2 ¼ x , where x0 is a characteristic lattice frequency, x20 ¼ k1 ðAAÞ , and I is the cormA x2 0
responding unit matrix. DðgA ; /y ; /z Þ is the dynamic matrix characteristic of the perfect fcc waveguide lattice and gA is a generic phase factor between two neighboring sites along the x-direction, gA ¼ ei/x ¼ eiqx a . The two variables /y and /z are the normalized dimensionless wave-vectors, defined as /y = qya and /z = qza. For simplicity we take a as the lattice distance between all adjacent sites, as well as on the boundary, neglecting any relaxation effects in the considered waveguides. The triplet (qx, qy, qz) correspond to the reciprocal lattice wave-vector components. Consequently, the doublet (/y, /z) (equivalent to (qy, qz)) describe the incidence angles of the system. The solution of Eq. (3) leads to three eigenmodes m = 1, 2, 3 with corresponding eigenfrequencies Xm and eigenvectors ~ um , at given wave-vector components ð/y ; /z Þ, for the perfect fcc lattice. We can distinguish two different kinds of solutions, namely the propagating elastic modes for which jgm j ¼ 1, and the physically evanescent modes for which jgm j 1. The evanescent modes are of interest because whilst they do not transport any energy they are nevertheless necessary for the detailed understanding of the scattering effects at the nanojunctions. To illustrate the model calculation, we present in Fig. 2a the dispersion curves, Xð/x Þ, for the elastic waves propagating, as a function of the normalized wavevector /x (/x ¼ qx a), where qx runs over the first Brillouin zone of the perfect fcc system. In
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229
3
mode 1 mode 2 mode 3
2.5
2
1.5
1
0.5
0 -4
-3
-2
-1
0
1
2
3
4
x Fig. 2a. Typical phonon dispersion curves for the perfect fcc crystallographic waveguide, presented over the first Brillouin zone.
this figure, all the phonon modes are acoustical; they are characterized by the limiting branch X ? 0 when /x ? 0. As may be seen, the eigenmodes labeled i 2 {1, 2, 3}, from bottom to top in that order, are propagating modes in the following frequency intervals: X1 X2 = [0.00, X1,max = 2.00], X3 [0.00, X3,max = 2.80]. It may also be shown that the eigenmodes of the fcc perfect waveguide are either symmetric or asymmetric with respect to the x-direction elastic waves propagation. We note that two of the three modes are degenerate. 2.2. Sandwiched system dynamics To analyze the phonons scattering in the presence of the interfacial region, as that in Fig. 1a and b, we not only have to know the propagating modes, but we have also to consider the evanescent eigenmode solutions of Eq. (3) as well. In other words, to determine the localized vibration states and the coherent phonons transmission via the nanojunctions in the sandwich systems, we use all the solutions gi , including the jgi j ¼ 1 propagating modes and the evanescent modes with jgi j 1. Both the propagating and evanescent eigenmodes can be obtained using different procedures [22,23]. The exact solutions are obtained as a function of the dimensionless frequency X, the wavevector ð/y ; /z Þ and the elastic interactions of the perfect fcc system. For an incoming elastic waves along the x-axis, in the eigenmode i, at given wave-vector ð/y ; /z Þ and frequency X, the scattering at the nanojunction boundary yields coherent reflected and transmitted elastic waves fields in the two semi-infinite A reservoirs. Given an incident elastic wave mode i from the left reservoir onto the interfacial region, the resulting Cartesian components a of the displacement field ~ u0a ðn; m; sÞ, for each fcc site, may be expressed using the matching approach [10–15], as the sum of the contributions coming from the incident elastic wave and from the fields backscattered at the nanojunction boundary, at the same frequency and wave-vector components
~ u0a ðn; m; sÞ ¼ gni~ ui þ
X
~ gn for n 0 j Rij uj ;
ð4Þ
j
ui and ~ uj denote the associated eigenvectors of the dynamic matrix for the perfect fcc system. The condition The vectors ~ n 6 0 refers the atomic layers before the interface domain. The notation Rij denotes the reflection coefficient which describe the scattering processes from the incident phonon mode i into the ensemble of the available eigenmodes j = 1, 2, 3 of the semi-infinite fcc perfect waveguide before the nanojunction domain. For sites inside the right hand reservoir A (semi-infinite lattice after the interfacial region), toward which the incident elastic wave mode i is transmitted, the resulting Cartesian components a of the displacement field ~ ua ðn; m; sÞ may be expressed by another appropriate superposition of the contributions from the forward scattered fields at the same the frequency and wave-vector components
~ ua ðn; m; sÞ ¼
X
gnj T ij~ uj ; for n d
ð5Þ
j
The condition n 6 d refers the atomic layers after the interface domain; d is the thickness of the interfacial region containing the sandwiched slabs. The notation Tij denotes the transmission coefficient which describe the scattering processes from the incident phonon mode i into the ensemble of the available eigenmodes j = 1, 2, 3 of the semi-infinite fcc perfect waveguide after the nanojunction domain. Let the vector j~ uint i regroup the atom displacements for the irreducible set of atomic sites on the interfacial region, where n 2 [0, d]. The atom dynamics equations for these sites coupled to those for neighboring sites in the matching domains of the
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reservoirs generate a rectangular system of equations with a number of unknown variables greater than the number of rows of this system. E Ei h T for the reflection and transmission proConsider next a Hilbert space constructed from the basis vectors j~ R ; ~ uint i; ~ uint i. The equations of motion coupled to the two reservoirs on both sides of the interface domain, may be cesses and from j~ E Ei h T . uint i; ~ R ; ~ written in terms of vector j~ Using the transformations connecting the displacement fields in Eqs. (4), (5), we obtain a square linear inhomogeneous system of the equations of the form
ih
h
E Ei
T uint i; ~ X2 I DðkAB ; /y ; /z Þ j~ R ; ~
! E ¼ IH ; gj ; /y ; /z
ð6Þ
! E Here the vector IH ; gj ; /y ; /z , mapped appropriately onto the basis vectors in the constructed Hilbert space and regroups the inhomogeneous terms describing the incoming elastic wave. DðkAB ; /y ; /z Þ is the matrix dynamic of the sandwich structure. Here kAB describes the elastic force interaction between sites of the two different lattices A and B. In the present analysis, by applying appropriate transformation, we obtain the matched atom dynamics matrix in the form
DðkAB ; /y ; /z Þ2121 ¼ M d ð21 27Þ MR ð27 21Þ for the A=B=A system and DðkAB ; /y ; /z Þ5151 ¼ M d ð51 57Þ MR ð57 51Þ for the A=B=A=B=A system; where Md and MR denote, respectively, the rectangular system of equations and the matching matrix. The solutions of Eq. (6) give explicitly the reflection, rij , and transmission, tij , coefficients in the studied sandwich structures, and yield consequently a complete description for the scattering processes at the nanojunctions for any given incident elastic wave. These are given by
8 2 < r ij ðX; /y ; /z Þ ¼ vv gi Rij gi : t ðX; / ; / Þ ¼ v gj T 2 ij y z v gi ij
ð7Þ
vgi is the group velocity of the eigenmodes i and Rij and Tij are, respectively, the reflection and the transmission probabilities. The scattering cross sections are normalized with respect to the group velocities of the elastic waves to ensure the unitarity of the scattering processes. For an evanescent phonon mode the corresponding group velocity is equal to zero. Typical results for the group velocities of the three eigenmodes are presented in Fig. 2b. Some modes show regions of anomalous dispersion where the group velocity is negative: these regions require a careful examination of the notion of forward and backward scattering for the elastic waves. The group velocity is related to the variation of the real wavenumbers accordingly to the frequency. But for imaginary wavenumbers, a null velocity is imposed [24]. In all cases the materials should have fully real modes. To determine this velocity, several methods can be used [25,26]. ! In this work, the group velocities are calculated directly from the slopes of the phonon dispersion curves, ~ v g ¼ gradXð~qx Þ. A challenge is encountered with this method at the locations where the dispersion branches cross. At these locations, it is difficult to keep track of which branch corresponds to each mode. Here, this challenge is avoided by following the approach of Wang et al. [27], and deriving an analytical expression for the phonon group velocity in terms of the dynamical matrix. This expression allows one to easily keep track of the frequency, group velocity, and polarization vector that correspond to a particular phonon mode. We can further define total reflection and transmission cross sections for a given eigenmode i, at given wave-vectors ð/y ; /z Þ and scattering frequency X, by summing over all the contributions of the scattered eigenmodes j, such that
X 8 r ðX; /y ; /z Þ ¼ r ij ðX; /y ; /z Þ > > < i j X > t ij ðX; /y ; /z Þ > : ti ðX; /y ; /z Þ ¼
ð8Þ
j
In order to describe the overall transmission and transmission cross sections of the sandwich systems at a given wave-vectors ð/y ; /z Þ and frequency X, it is useful to define the transmittance of the systems rðX; /y ; /z Þ, by summing over all input and output propagating phonon modes
rðX; /y ; /z Þ ¼
XX i
tij ðX; /y ; /z Þ
j
where the sum is carried out over all propagating phonon modes at frequency X and given wave-vectors ð/y ; /z Þ.
ð9Þ
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231
12
10 Vg 1 Vg 2 Vg 3
Vg
8
6
4
2
0
0
0.5
1
1.5
2
2.5
3
Fig. 2b. Curves of the group velocities of the different propagating modes (1–3) of the perfect fcc crystallographic waveguide Vg, as function of the scattering frequency X.
3. Numerical applications and discussion The scattering of the phonons at the interface between two similar fcc lattices is considered, in this work, with reference to elastic waves incident from the left reservoir to the right reservoir of the scattering region in Fig. 1. We examined two sandwich structures containing different number of atomic layers in the interfacial region. They are modeled respectively in Fig. 1a and b, where the thickness of the interfacial domain is defined by the distance d. For the two studied systems, the wavevector representation can be used in the transverse direction due to its infinite size along the interface domain [28]. The typical geometry of the first system A/B/A is presented in Fig. 1a. On the other hand, Fig. 1b gives the typical geometry of the second system A/B/A/B/A. In the two forms of the interface domain, we have simulated the case where each sandwiched slab is composed by five atomic layers. When the elastic wave arrives at the nanojunction zone, the fist one is split into its transmitted and reflected parts. Consequently, the nanojunction zones joining different types of atomic slabs are considered as effective atomic inhomogeneities in the studied structures. In this work, the model is applied to simulated the phonon scattering in the two considered fcc sandwich structures, by using Eqs. (6)–(9). The dynamic properties, the phonon transmission/reflection, localized vibration states and the total transmittance are calculated numerically only for comparable masses (mA mB). In addition, the analysis is carried out, also, for three different cases of the interface elastic constants kAB, where each case determines a choice of the elastic properties of the interface domain. These are defined, respectively, by: (i) kAB = 0.75, (ii) kAB = 1.00, (iii) kAB = 1.25. We remind that the interface elastic constant kAB is considered as the arithmetic mean of the A and B nearest neighbor interaction constants, so then kAB = (kAA + kBB)/2. The first case (i) corresponds to a softening of its elastic constants in comparison to the bulk of the lattice A; the second case (ii) corresponds to the situation where the elastic constants are the same, and in contrast the third case (iii) corresponds to a hardening of its elastic constants. These possibilities are similar to situations where changes in elastic constants may take place in the neighborhood of steps and kinks in surfaces [29]. The purpose of this numerical procedure is to investigate how the local dynamics respond to local changes in the elastic properties. To study the phonon transmission and reflection effects, the inhomogeneous Eq. (6) is solved numerically. The transmission spectra are calculated from Eq. (7), for a given incident angle ð/y ; /z Þ ¼ ð0; 0Þ. The results are presented in Fig. 3a for the system A/B/A and in Fig. 3b for the second system A/B/A/B/A. In Fig. 3, we observe that the transmission and reflection coefficients are presented as a function of the scattering frequency X and at a given direction for the three phonon modes (t1, r1), (t2, r2) and (t3, r3). The results are obtained for the three cases of elastic constants: rd = 0.75, 1.00 and 1.25. In all modes, we show that the transmission and the reflection coefficients verify the unitarity condition s = ti + ri = 1, (i = 1, 2, 3) of the scattering matrix and that this is used as a check on the numerical calculation. We note that all transmission coefficients are hopeless for X superior to 2.80. As expected the influence of the sandwiched films is nil at low frequencies, where X tends to zero, the transmission coefficients t1, t2 and t3, tend to her maximum values (1); in contrast, at the high frequency the coefficients are sensitive to the effect of the interface and become zero when X tends to 2.00 for the modes 1 and 2, and X tends to 2.80 for the mode 3. Another general characteristic of the phonon transmission is the displacement of its spectral features to higher frequencies with increasing hardening of the elastic force constant kAB between the A and B layers, for the two configurations, which is a signature of the nanojunction matching necessary for the transmission the elastic waves.
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Transmission and reflection coefficients
1.5
t3 r3 s3
1
1.5 t2 r2 s2
1
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0 0
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t1 r1 s1
1 0.5
0 0
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0 0
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0 0
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3 t1 r1 s1
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1
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2
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3 t1 r1 s1
0
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1
1.5
2
2.5
3
Fig. 3a. Phonon transmission t1, t2, t3 (dotted lines) and reflection r1, r2, r3 (solid lines) coefficients and their sum s1, s2, s3 (dashed lines) across a single sandwich slab in the fcc structure A/B/A. They are given as function of the scattering frequency X, at a fixed vector components (/y, /z) = (0, 0).
1.5
Transmission and reflection coefficients
1.5
t3 r3 s3
1 0.5
0.5
0
0
0
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1
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2
2.5
3
0
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t2 r2 s2
1
0.5 0.5
1
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2
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3
0
0
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1
1
1
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0
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1
1
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0
0
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1
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2
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0
t1 r1 s1
1
0.5
1
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2
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3
0
0
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1
1.5
2
2.5
3 t1 r1 s1
0.5
1
1.5
2
2.5
3 t1 r1 s1
0.5
1
1.5
2
2.5
3
Fig. 3b. Phonon transmission t1, t2, t3 (dotted lines) and reflection r1, r2, r3 (solid lines) coefficients and their sum s1, s2, s3 (dashed lines) across a triple sandwich slabs in the fcc structure A/B/A/B/A. They are given as function of the scattering frequency X, at a fixed vector components (/y, /z) = (0, 0).
Some of the spectral maxima in the interval X 2 [0, 2.80] correspond to characteristic Fano resonances, since the atom vibration states on the interfacial domain are effectively localized states embedded in the continuum of the waveguide incident phonon modes. In another words, this is attributed to the coupling of the propagating incident elastic waves with a localized vibration modes induced by the interface domain. However, we can conclude that the waveguide configured in Fig. 1b introduce more localized modes, i.e., more Fano resonances in transmission spectra than the structure configured in Fig. 1a. In addition, we observe that the interfacial phonon transmission decreases with increasing the thickness of the sandwiched films, which is well expected due to the multiple interface effects. We note that when phonons are transported in multilayered structures, the elastic waves can interfere with each other to form mini phonon bands and can thus change the phonon transmission. Moreover, similar transmission behavior has been observed for phonon radiative transport across periodic structures [30]. In Fig. 4, we plot the total phonon transmission rðX; /y ; /z Þ (or the phonon transmittance) across the interface domain, given by formula (9) for the direction ð/y ; /z Þ ¼ ð0; 0Þ. For the first sandwich system A/B/A, the phonon transmittance conductance is presented, in Fig. 4a, throughout the interval X 2 [0, 2.80], which corresponds to the interval frequency of propagation phonon modes from the first semi-infinite reservoir A to the second reservoir A after the insured thin film B. In the other hand, in Fig. 4b, we present the total phonon transmission via the interfacial domain of the system A/B/A/B/A, which are propagating also in the frequency interval X 2 [0, 2.80]. In the two cases of the sandwiched thin films examined in this work, the conductance spectra always starts with her higher values at low frequencies (X tend zero) and decrease with increasing X then becomes zero at the Brillouin Zone limit. The curves of the phononic transmittance present the pick resonances of different height and widths, what gives him a rough aspect.
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233
3.5
3
2.5
2
1.5
1
softening homogeneous hardening
0.5
0
0
0.5
1
1.5
2
2.5
3
Fig. 4a. The curves of the phonon transmittance for the fcc sandwich structure A/B/A, as function of the scattering frequency X, at the given direction (/y, /z) = (0, 0).
3.5
3
2.5
2
1.5
1 softening homogeneous hardening
0.5
0
0
0.5
1
1.5
2
2.5
3
Fig. 4b. The curves of the phonon transmittance for the fcc sandwich structure A/B/A/B/A, as function of the scattering frequency X, at the given direction (/y, /z) = (0, 0).
The phonon transmittance, rðX; /y ; /z Þ, is less than or equal to one phonon throughout the frequency interval propagation, for all two examined configurations. This illustrates how the sandwiched thin films at the heart of the nanojunction domain in each case constricts the transmission to a maximum of one phonon at a time, which is characteristic for the atomic layers with finite thickness or confined films n 2 [0, d]. In all cases, the conductance spectra always starts with her higher values at low frequencies and fluctuate for the intermediate values of X and decrease with increasing X then becomes zero at the Brillouin Zone limit. In Fig. 4a, the phonon transmittance via the atomic interface in the first system A/B/A is more important than the phonon transmittance in the second system A/B/A/B/A given in Fig. 4b; this can be explained by thickness of the sandwiched thin films and the velocity of decay of the corresponding wave-function which is different in the both sides of the interfaces domain. In each figure, we observe the systematic displacement of its spectral maxima to great values; this can be explained by the difference value of the elastic constant parameter kAB, and a situation that confirms the importance of the interface domain localized states in the transmittance spectra. Another interesting observation is the important magnon transmittance, rðX; /y ; /z Þ, in the interval X 2 [0, 2.00], for the two studied systems, and for all considered elastic constant kAB. The observed maxima in the interval mentioned interval frequency correspond to characteristic Fano resonances since the vibration states on the interface domain are effectively localized discrete states which interact with the continuum of the incident phonon waveguide modes. In particular, for the first system A/B/A, the features of the conductance spectrum attributed to Fano resonances, XF, correspond to the
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maxima that shift through the frequencies with increasing kAB. In each mode, the observed Fano resonance positions XF = 1.36, 1.74 (modes 1 and 2) and XF = 1.95, 2.45 (mode 3), for the softening elastic constant situation (case (i)) shift to the following higher frequencies XF = 1.44, 1.82 (modes 1 and 2) and XF = 2.05, 2.58 (mode 3) for the hardening situation, which corresponds to the case (iii). The quoted remarks are also valid for the second system A/B/A/B/A. The features of the Fano resonances are observed at the positions XF = 0.88, 1.38, 1.74 (modes 1 and 2) and XF = 1.26, 1.94, 2.45 (mode 3), for the case (i). With increasing kAB (case (iii)), XF = 0.94, 1.45, 1.82 (modes 1 and 2) and XF = 1.33, 2.05, 2.58 (mode 3). For the case (ii), which describes the situation where the elastic constants are same everywhere, a total phonon transmission are observed. In this case, the considered systems correspond to perfect lattices without inhomogeneities. We have also applied the model to calculate numerically the energies of the localized vibration states in the neighborhood of the interface zone between the two semi-infinite reservoirs A. These energies are obtained by resolution of the determinant det½DðkAB ; /y ; /z Þ ¼ 0 of the matrix mentioned in Eq. (6). They correspond to propagating phonons via the nanojunction zone, which are spatially localized in the sense that their atom vibration amplitudes decay along the x-direction inside the two semi-infinite reservoirs located on both sides of the interfacial domain. These localized vibration modes are induced by the breakdown of the translation symmetry engendered by the interface zone. Moreover, the localized modes couple with the incident propagating phonon modes in the scattering region. These localized states are presented in Figs. 5a and 5b in the form of bulk displacements for the perfect lattice and the dispersion curves of the irreducible sites of the interfacial region. These dispersion branches are given as a function /y, propagating parallel to the interface plane nanojunction. The purpose, of the calculations, of the elastic wave dispersion curves for A bulk and interface modes, is to demonstrate the essential features of the elastic wave interface modes and the influence of the parameter kAB on the localized modes of phonons at the interface zone. As has been pointed out in Section 2, where jgj ¼ 1 corresponds to the propagating modes in the semi-infinite A perfect regions as shown in the shaded area in Figs. 5a and 5b. Near the interfacial zone, the dispersion curves depict phonons propagating along the x-direction (normal to the interface) that are, however, effectively localized in the sense that their amplitude of the displacement is evanescent in the direction normal to the surfaces of the two semi-infinite reservoirs A, located on both sides of the interface domain. The amplitude of the localized elastic wave in the interface zone decays exponentially with increasing penetration into A subsystems. In Fig. 5b, the interface dispersion branches contains the dispersion branches of Fig. 5a, this can be explained by the fact that the effects of the sandwiched films are superimpozed. In addition, the localized states displace to higher energies with increasing elastic constant between the atomic of the two lattices A and B. We note that the number of the localized vibration states of the first system A/B/A is different from those of the second system A/B/A/B/A. This is explained by the increasing thickness of the sandwiched layers between the A reservoirs. Varying the parameters of the boundary domain can technically control the elastic waves transmission properties. This implies the possibility of a nanometric procedure to organize the heat transfer from a thermal or coherent source, preferentially, into different branches of a mesoscopic system.
3.5 softening case hardening case
3
2.5
2
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1
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0
0
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1
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2
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3
3.5
y
Fig. 5a. Curves of the localized vibration sates on the boundary of the fcc sandwiched A/B/A system. The A bulk energies are shown shaded, whereas the circular symbols represent the interface modes.
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235
3.5 softening case hardening case
3
2.5
2
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1
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0
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y
Fig. 5b. Curves of the localized vibration sates on the boundary of the fcc sandwiched A/B/A/B/A system. The A bulk energies are shown shaded, whereas the circular symbols represent the interface modes.
4. Conclusion In conclusion, we have studied the generic properties of phonon propagation and elastic waves diffusion by different atomic nanojunctions in cubic lattices by the matching technique in the harmonic approximation. Our results show that the phonon transmission across sandwiched ultrathin films in fcc lattices can be controlled by the thickness of the inserted films. Additionally, the phononic transmittance of the fcc sandwiched systems is a strong function of the scattering frequency, vector components and the atomic configuration. The formalism presented here is independent of the configuration of the interface region, which makes it easy to extend them to study a variety of disordered interface problems. At lower frequencies the individual and total transmission spectra start with its maximal values, present the fluctuation for 0 6 X 6 2.80 and become null at the Brillouin Zone limit. The fluctuations number related to Fano resonances are in good agreement with the localization theory. Our work demonstrates that the number of localized modes which interact with the propagating modes of the continuum is proportional to the number of the sandwiched films in the interfacial zone. This explains the higher number of fluctuations in the vibration spectra of the system with triple sandwich slabs. Acknowledgment The authors would like to thank Dr. Louiza Belkacemi from the University of Houston, TX, USA for editing our manuscript. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
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