Surface Wave Propagation in Thin Silver and Aluminium Films Anouar Njeha, Nabil Abdelmoulab, Hartmut Fuessc, and Mohamed H´edi Ben Ghozlenb a Institut Pr´eparatoire aux Etudes d’Ing´enieurs de Sfax, BP 805, 3018 Sfax, Tunisia b Laboratoire de Physique de Mat´eriaux, Facult´e des Sciences, Universit´e de Sfax, 3018 Sfax, Tunisia c Institute of Materials Science, University of Technology, Petersenstr. 23, D-64287 Darmstadt, Germany Reprint requests to Dr. A. N.; Fax: +216-74-243542; E-mail: [email protected] Z. Naturforsch. 60a, 789 – 796 (2005); received September 16, 2005 Three kinds of acoustic waves are known: bulk waves, pseudo-surface waves and surface waves. A plane wave section of a constant-frequency surface of a film serves as a hint for the expected nature. Calculations based on slowness curves of films reveal frequency ranges where each type of acoustic waves is predominant. Dispersion curves and displacement acoustic waves are calculated and commented in each frequency interval for different coated materials. Both dispersion and sagittal elliptical displacement are sensitive and depend on diagrams mentioned above. Silver and aluminium thin films having different anisotropy ratios, namely 2.91 and 1.21, are retained for illustration. Key words: Surface Acoustic Waves; Propagation; Dispersion; Thin Films; Mechanical Properties. 1. Introduction 2. Surface Acoustic Wave Analysis The preparation of solid films requires the under- An ultrasonic surface wave propagates at the surface standing of their mechanical properties. Such films are of a homogenous material with an amplitude that de- parts of microelectronic devices and improve the hard- creases exponentially perpendicular to the surface and ness and wear resistance of coatings. vanishes to negligible values within a few wavelengths Surface acoustic waves (SAWs) are convenient tools below the surface [8]. The penetration depth decreases to study the mechanical properties of thin films, be- with increasing frequency. The velocity of propagation cause the rate of their energy away from the sur- is somewhat smaller than the bulk shear velocity as- face is frequency-dependent. With increasing SAW sociated with the material, which is the same for all frequency, the energy is concentrated closer into the frequencies. If the material is coated with a film which surface. Therefore the influence of the elastic proper- has different elastic parameters, the surface wave will ties of the film on the velocity increases as well [1 – 4]. become dispersive. In coated material the velocity of SAWs depends on We are going to concentrate on the acoustic proper- their frequency, i. e. shows dispersion. The wavelength, ties of an anisotropic solid layer of thickness h, which frequency and thickness of the layer are varied to ob- is perfectly bound to a semi-infinite anisotropic sub- tain dispersion curves of the SAW. The elastic proper- strate. ties are then derived from these dispersion curves [5]. We use the coordinate system shown in Fig. 1, with In anisotropic media the speed of acoustic waves the x1 and x2 axis lying in the interface between the varies with the propagation direction. Particularly, di- layer and the substrate, the x3 axis being directed into rections exist where the lowest speed of a quasi- the substrate. Here, the plane with x3 = 0 is the in- transverse bulk wave approaches the speed of terface between the layer and the substrate, while the SAWs [6, 7]. plane with x3 = −h is the free surface. The ultrasonic The purpose of this paper is to describe the depen- surface wave propagates along the x1 axis. The wave dence of SAWs in Ag/Si and Al/Si systems on the must satisfy the appropriate wave equation in the layer propagation direction and the anisotropy of the mate- and the substrate, and the boundary conditions im- rial. Special attention is given to SAW displacements: posed by the interface and the free surface. The wave The analysis includes mainly its elliptical characteris- equation for the particle displacement in a perfectly tics. elastic and homogeneous medium can be written, in 0932–0784 / 05 / 1100–0789 $ 06.00 c 2005 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen · http://znaturforsch.com 790 A. Njeh et al. · Surface Wave Propagation in Thin Ag and Al Films Writing the displacements as linear combinations of terms having the phase velocity ν, the displacement field in the layer is given as [7, 10]: (n) ˜ (n) u˜ j = ∑ A˜nu˜ exp(ikβ x3) exp[ik(x1 − νt)], n 0 j (5) n = 1to6, and in the substrate as: = (m) ( β (m) ) [ ( −ν )], u j ∑m Amu0 j exp ik x3 exp ik x1 t Fig. 1. Coordinate system for surface acoustic waves propa- gation in a thin film. m = 1,3,5. (6) the absence of body forces and piezoelectric effects, The coefficients Am with even subscript m are omit- as [7] ted because they are not related to finite solutions. ( ) ( ) u˜ n and u m are the complex eigenvectors associated 2 2 0 j 0 j ∂ ui ∂ uk ρ = C , (i, j,k,l = 1,2,3), (1) with (3) for the film and for the substrate. A˜n and Am ∂ 2 ijkl∂ ∂ t x j xl are the weighting factors for the linear combination of waves in the film (5) and the substrate (6). Only in which ρ is the density of the medium and C is the ijkl β (n) (m) elastic tensor. The components in (1) refer to the coor- one eigenvector exists for each root .˜u 0 j and u0 j dinate system of Fig. 1, and the summation convention are the components of the eigenvector corresponding ρν2 ρν2 on repeated subscripts is implied. to the eigen values ˜ and , respectively. The In the following, a quantity carrying a tilde desig- weighting factors must be determined by the applica- nates a quantity of the film and the same quantity with- tion of the boundary conditions based on the displace- ments in (5) and (6). The particle displacements and out a tilde refers to the substrate. Thus, ρ˜ and C˜ ijkl are the density and the elastic tensor of the layer mate- the traction components caused by the stress compo- nents of the wave (T , T and T ) must be contin- rial, whereas ρ and Cijkl are the density and the elastic 13 23 33 tensor of the substrate. We assume a straight crested uous across the interface under the assumption of a rigid contact between the two materials. Since the free surface wave propagating in the x1 direction with surface is considered to be mechanically stressed free, u j = u0 j exp(ikβx3)exp[i(kx1 − ωt)], (2) the three traction components of stress must vanish thereon, and nine boundary conditions are obtained. In where the components of the wave vector k(k1 = order to obtain nontrivial solutions of this set of ho- k,k2 = 0,k3 = kβ), the angular frequency ω and the mogenous equations, the 9 · 9 determinant of the coef- components of the polarisation vector u 0 are related ficients A˜n and Am must vanish [7]. through the set of three linear equations [9] 2 3. Results (Cijklkikl −ρω δ jk)u0 j = 0, i, j,k,l = 1,2,3, (3) In this paper cubic symmetry of the film is as- for which the secular equation is sumed. The substrate is a single-crystal Si(001) and 2 x =[ ] |Cijklkikl − ρω δ jk| = 0, i, j,k,l = 1,2,3. (4) the wave vector is defined as 1 100 . We con- sider both a thin silver and a thin aluminum film of For a given k and ω (4) is of the order 6 in β and may 200 nm thickness, and two different wave propaga- be solved for the six roots β (n) (n = 1 to 6). The six tion directions are investigated. These are [100], the roots for β are either real or occur in complex con- Xp1 axis, and [110], the Xp2 axis. For both con- jugate pairs. All six roots are used for the calculation figurations, the (001) plane for the film corresponds of the film displacement, but only the three roots with to the interface film/substrate, and two configurations positive imaginary parts are taken for the substrate dis- Ag[100]/Si[100] and Ag[110]/Si[100] are analyzed for placement. an SAW excited along [100] in the substrate. A. Njeh et al. · Surface Wave Propagation in Thin Ag and Al Films 791 Fig. 2. Surface acoustic wave (SAW) velocity of the first Rayleigh mode in an Ag/Si(001) layered system as a function of the product kh, where k is wave number and h is thickness of the film: (a) the wave propagates along the Xp1 =[100] axis into the Ag film; (b) the wave propagates parallel to the Xp2 =[110] axis into the Ag film. Table 1. Anisotropic materials parameters. for film and substrate. In a second step the boundary ρ 3 conditions determinant is evaluated. The phase veloc- Material C11 [GPa] C12 [GPa] C44[GPa] [kg/m ] ν Silicon 165.5 63.9 79.5 2329 ity is iteratively varied until this determinant ap- Aluminum 107.3 60.8 28.3 2702 proaches zero. Repeated for each frequency, the SAW Silver 123 92 45.3 10500 dispersion curves for the thin film coated crystal are obtained. The determination of the dispersive curves was In Fig. 2, the SAW velocities of the first Rayleigh based on the elastic characteristics reported in Ta- mode, calculated at various frequencies for an Ag film ble 1 [11]. grown on the (001) plane of the Si substrate, are dis- played as functions of kh.