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Effect of roughness on the phase velocity of Rayleigh waves in GaAs crystals
Thin Solid Films 491 (2005) 184 – 189 www.elsevier.com/locate/tsf
Effect of roughness on the phase velocity of Rayleigh waves in GaAs crystals N. Tarasenko, P. Bohac, L. Jastrabik, D. Chvostova, A. Tarasenko * Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21, Prague 8, Czech Republic Received 7 July 2004; received in revised form 25 May 2005; accepted 25 May 2005 Available online 11 July 2005
Abstract The phase velocity dispersion of the Rayleigh waves on surfaces of GaAs crystals having different roughness has been measured by laseracoustic wave equipment. Applied model of a rough film with zero elastic moduli explained the effect of roughness on the acoustic surface wave phase velocity and showed good coincidence with the experimental data. D 2005 Elsevier B.V. All rights reserved. Keywords: Gallium arsenide; Surface roughness; Surface acoustic waves; Phase velocity
1. Introduction A well-known phenomenon of elastic energy propagation along the free surface of a solid of infinite depth is the Rayleigh waves. The waves propagate in directions parallel to the surface only. The amplitude of the waves decays exponentially in direction perpendicular to the surface. Therefore, the wave energy is concentrated near the surface in a narrow layer within a few wavelengths. The waves are localized near the surface of the substrate—the surface acoustic waves (SAWs). Rayleigh waves have attracted considerable theoretical and experimental interest during past decades because of their practical utility in many applications. The SAWs can be also used for investigations of solid surfaces and films, deposited on the surfaces [1]. The damage layers and films adsorbed on the substrate surface cause dependence of the SAW phase velocity on the wave frequency, which is called dispersion. The dispersion depends on the elastic parameters of the layer as well as on the layer density and its thickness. Interpretation of measurements is based on the general theory of elastic waves, which is well developed for the most important cases (see, for example, Ref. [2]). A system * Corresponding author. Tel.: +42 266052165; fax: +42 286581448. E-mail address: [email protected] (A. Tarasenko). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.05.049
of the wave equations for the elastic displacements in the substrate crystal and the layer should be solved and solutions have to satisfy the boundary conditions for the displacements and stresses at the surface and interface between the substrate and layer. It should be noted that the real systems have not such ideal surfaces and interfaces as are supposed in the theory. Typically, the surfaces are far from the perfect flat planes. The surface roughness can be detected in varying degrees on all solid surfaces. The surface roughness leads to the scattering and attenuation of the SAWs. There are some theoretical and experimental investigations of this effect [3 – 6]. It should be noted that the same effect of the interface roughness on Lamb waves propagation have been investigated in Ref. [7]. We have proposed a simple approach for estimation of the phase velocity dispersion caused by the surface roughness in another limiting case when the rough surface is described by an extremely sharp profile function [8]. Using this approach we considered propagation of the surface elastic waves in crystals having cubic symmetry and obtain equations, determining the SAW dispersion for the substrate with rough surface. It was shown that the rough surfaces give rise to the dispersion of the Rayleigh mode. The results are compared with the experimental data of the SAW phase velocity obtained on the GaAs samples. We obtained a
N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
rather good coincidence between the experimental data and calculated curves by fitting only one parameter—the effective thickness of the rough layer proportional to the rough layer density. There is a good correlation between the fitted values of the effective thickness and the measured RMS value of the surface roughness.
2. Model and boundary conditions Let us consider a semi-infinite substrate bounded by a rough surface (see Fig. 1). The substrate fills the half-space x3 fðY r Þ and the rough surface is determined by the profile random function x3 ¼ fðY r Þ. The theory of elastic waves propagation in the solid media and layered systems is well established. We mention here the milestones of the method only, referring the interested readers to the excellent reviews for detailed descriptions [2,9 – 13]. The elastic displacements must satisfy the wave equations in the substrate q
B 2 uj B2 um cijmn ¼ 0; 2 Bt Bxi Bxn
ð1Þ
where u i is the component of the displacement along the coordinate axis x i ; c ijmn are the components of the corresponding elastic stiffness tensor; i, j, m, n = 1, 2, 3. The elastic stiffness tensor c ijmn in Eq. (1) is referred to the coordinate system of Fig. 1 and the summation convention on repeated indices is implied. The solutions of the system are assumed to be in the following form Y uY ðY r ; t Þ ¼ aY expðakx3 Þexp ik Y r vt : ð2Þ They describe plane waves propagating along a direction Y determined by the wave vector k with a phase velocity v. The displacements have constant phase and amplitude along Y any line perpendicular to the vector k . The amplitude of the displacements decays exponentially in the depth of the substrate (the real part of a must be positive only). The rate of decay is chosen as ak in order to simplify formulae.
185
Due to the symmetry properties of the stiffness tensor, its indices are collapsed by pairs as follows: 11Y1, 22Y2, 33Y3, 23, 32Y4, 13, 31Y5, 12, 21Y6 (see, for example, Ref. [14]). In the case of the GaAS substrate, the only nonzero components of its stiffness tensor c ijmn are the following: c 11 = c 22 = c 33, c 12 = c 13 = c 23 and c 44 = c 55 = c 66 m (c 11 c 12) / 2. Then the solution of the system Eq. (1) is reduced to the secular sixth-order algebraic equation in the quantity a. There are six roots a j , j = 1, 2,. . ., 6. Each root a j has a corresponding eigenvector aY ð jÞ. Any partial wave (solution with a definite a j and aY ð jÞ ) satisfies the equations of motion. The general solution will be a linear combination of all partial waves uY ¼
6 X
Y Cj aY ð jÞ exp aj kx3 exp ik Y r vt :
ð3Þ
j¼1
One must choose the non-vanishing values only for C j ’s, which correspond to roots lying within the right half of the complex plane (real part of a j must be positive), to ensure decaying of SAWs with the depth of the substrate, x 3 0. There are only three of them satisfying this condition. In the [110] and [100] directions of the SAW propagation the secular equation splits into second-order and forth-order multipliers. For the [110] direction it has very simple form q þ w a2 a4 2Pa2 þ Q ¼ 0; ð4Þ where: i 1 h wð1 þ rÞ þ rðr þ sÞ=2 ð1 þ sÞ2 ; 2r w Q ¼ ½w þ ðr þ sÞ=2; r P¼
ð5Þ
and other quantities are defined as follows: r = c 11 / c 44, s = c 12 / c 44, q = (r s) / 2 1, w = 1 (v / v t)2, v t = (c 44 / q)1/2 is the transverse shear velocity. One can determine easily the roots and eigenvectors a1;2 ¼ aY ð1;2Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PF 0
P2 Q;
pffiffiffi 1 ið1 þ sÞa1;2 =p2ffiffiffi ¼ @ ið1 þ sÞa1;2 = 2 A; w þ ðr þ sÞ=2 a21;2
a3 ¼ aY ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffi q þ w; 0 1 1 B C ¼ @ 1 A: 0 ð6Þ
Fig. 1. Schematic view of the substrate with the rough surface (crosssection in the plane x 2 = 0) and the system of coordinates assumed for the (001)-oriented surface.
The investigations of the propagation of SAWs on the rough surfaces were carried out on the basis of Rayleigh method [15]. In this method the components of the elastic displacement field are expanded in terms of solutions of the equations of motion of the substrate in the region x 3 0. The coefficients in these expansions are determined by continuing the expansions into the rough region 0Vx3 VfðY r Þ, and satisfying the stress-free boundary
186
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conditions at the surface x3 ¼ fðY r Þ. This leads to the integral eigenvalue equation for the displacement field. The problem is rather complex and only recently a few attempts have been made to investigate rough surfaces of an anisotropic medium [16 – 18]. The Rayleigh approach is valid only when the surface profile is sufficiently smooth. It can not be applied for the profiles consisting of the sharp, abrupt, narrow peaks and valleys as shown in Fig. 1. If the characteristic size of the surfaces inhomogeneities is less than the Rayleigh wavelength k, the SAW can not Fenter_ the rough region. One can imagine the rough surface as an arrangement of microparticles (blocks) of different shape and size, randomly scattered over an ideal flat reference plane, x3 ¼ 0ufmin ðY r Þ, which separates the ideal elastic media, x 3 < 0, and the rough, discontinuous part, 0Vx3 VfðY r Þ. The blocks are independent, there is no lateral interaction between them. Any block moves in accordance with the surface local displacements, but the block displacement do not influence the displacements of the nearest neighboring blocks. It is clear that the elastic waves can not propagate through such Fgranular_, discontinuous medium. Nevertheless, it should influence the propagation of the SAW in the underlying host crystal. The model can be applied to describe the mechanical properties of the island films. Such films are composed of the well-separated particles which do not contact each other. The sizes of these particles vary in wide region depending on the film deposition conditions. Damage layers can be another field of the model application. Surfaces of wafers, sliced by diamond saws from ingots have rather high density of lattice defects, cracks, etc. Such discontinuous, highly porous layers of varying thickness influence also the propagation of SAW. If k exceeds the characteristic size of the blocks, all specific effects related with the shape of the blocks and their geometrical arrangement are smoothed out. As the rough layer cannot support the propagation of the elastic waves one should set the elastic moduli of the medium equal to zero. The only physical property of the layer is the surface density of the blocks, .. It equals to the mass of the blocks covering 1 cm2 of the surface: qˆ .¼ S
Z
fðY r Þdx1 dx2 ;
ð7Þ
S
where S is the area of the sample surface and qˆ denotes the ordinary bulk density of the blocks. The presence of such layer changes the boundary conditions on a flat surface. Instead of the stress-free boundary conditions on the rough surface one has the boundary conditions on a flat surface loaded by the rough layer. To account for the presence of the rough layer we will define the boundary conditions not at the real surface, but at the plane x 3 = 0, which separates the rough layer from the continuous elastic part. The rough layer moves in accord-
ance with the plane displacements. Therefore, the stresses T 3j arising in the plane x 3 = 0 are equal to the inertia forces, due to the acceleration of the rough layer T3j ¼ .
B2 uj at x3 ¼ 0; Bt 2
j ¼ 1; 2; 3:
ð8Þ
It should be noted that these boundary conditions coincide with the ordinary conditions T 3j = 0, if the surface density of the blocks, . = 0, i.e. when the roughness is absent and the outer surface of the film, x 3 = 0, is an ideal flat plane. This approximation reduces the problem of rough surfaces to the simple case of the SAW propagation in the structures, having ideal surfaces, by modification of the boundary conditions. Amplitudes C j ’s must be determined from the boundary conditions Eq. (8), which can be written as follows. Bu1 Bu3 þ c44 ¼ .ðkvÞ2 u1 ; Bx3 Bx1 c44
c11
Bu2 Bu3 þ Bx3 Bx2
¼ .ðkvÞ2 u2 ;
Bu3 Bu1 Bu2 þ c12 þ ¼ .ðkvÞ2 u3 : Bx3 Bx1 Bx2
ð9Þ
In order to have nontrivial solutions, the determinant of the system Eq. (9) must be set to zero: ða hÞað1Þ þ iað1Þ =pffiffi2ffi 1 1 3 ða hÞað1Þ þ iað1Þ =pffiffi2ffi 1 1 3 ðra1 hÞað1Þ þ ispffiffi2ffiað1Þ 3 1
ð2Þ ð2Þ pffiffiffi ða2 hÞa1 þ ia3 = 2 ð2Þ ð2Þ pffiffiffi ða2 hÞa1 þ ia3 = 2 pffiffiffi ð2Þ ð2Þ ðra2 hÞa3 þ is 2a1
a3 h h a3 ¼ 0: 0
The rough layer changes the dispersion equation of SAW. The phase velocity depends on the angular frequency, x = kv. The effect of the rough layer is determined by a small parameter, h = xvd/v2t b 1, where d = ./q is the effective thickness of the damage layer.
3. Quantitative surface evaluations We characterize the surface roughness by ordinary statistic parameters that represented the average properties of the random profiles R a and R q. The R a is the center-line average (CLA) roughness: Z 1 Ra ¼ jfðY r Þ fa jd x1 ; ð10Þ L L and R q is the root mean square (RMS) roughness. It obtained as an average of the squared values of the measured height deviations fðY r Þ fa , then taking the square root of the mean value
N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 2 Rq ¼ ðfðY r Þ fa Þ dx1 : L L
ð11Þ
Here L is the length of the scan and f a is the position of the mean line, which is defined as follows: the total area of the material-filled profile above the mean line should be equal to the total area of voids below the mean line. It follows from the definition, that 1 fa ¼ L
Z
fðY r Þdx1 :
ð12Þ
L
For one-dimensional profiles f a = d(q/qˆ). It coincides with the effective thickness of the rough layer d, if q = qˆ, but may be substantially different in the case of island films. For two-dimensional rough layers the quantities f a and d are different by their definitions. The position of the center-line and the roughness parameters R a and R q depend on the height distribution along some direction, but d depends also on the three-dimensional shapes of the peaks and valleys of the rough layer.
4. Experimental details The non-destructive Laser-Acoustic Wave Analyzer made in the Fraunhofer Institute for Material and Beam Technology (Dresden, Germany) was used for measurements of roughness influence on the SAWs propagation. The analysis is based on measurements of phase velocity of the SAW and yields the dispersion curve. The experimental setup is shown in Fig. 2. The Analyzer is equipped with N 2 laser (wavelength 337 nm). A cylindrical lens focuses the laser beam in a 5 mm abscissa on the
187
sample surface. Short laser pulses with energy 400 AJ and duration 0.5 ns generate SAW impulses in wide band spectrum in tens MHz range. The SAW impulses are detected by a piezoelectric transducer using a PVDF foil pressed by a steel wedge to the sample surface. The acquisition of the electrical signals is performed by a digital oscilloscope with sampling rate of 2 GSa/s (bandwidth 500 MHz). The oscilloscope is triggered by photodiode signal generated by the laser pulse. PC performs the signal processing and controls the measuring procedure. The sample and wedge transducer are fixed to a translation stage driven by a computer controlled DC motor with an accuracy Dx = T2 Am. Both move perpendicular to the laser beam to vary the distance between focused laser spot and transducer. The smoothed velocity spectrum is evaluated from one measurement package commonly consist of these 6 stepwise abbreviating distances. The dispersion curve is calculated automatically using the selected range of the velocity spectrum. The software of the Analyzer is based on the solution of the inverse problem of the dispersion of Rayleigh waves on a homogeneous isotropic substrate covered by an isotropic layer [19,20]. Very small variations of the phase velocity can be investigated due to the rather high accuracy of the method. The measuring error of the phase velocity is less than T 1 m/s. The details of the photoacoustic method can be found elsewhere [21,22]. For experimental investigations of the influence of surface roughness on propagation of the Rayleigh mode we have used two wafers sliced from a (001)-oriented GaAs single crystal. The wafer size was <30 2.6 mm. Initially, one of the wafers was rough ground by SiC powder and the latter has polished surface. After each measurement the initially polished surface of the sample was stepwise ground
Fig. 2. Schematic representation of the measuring equipment [21].
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N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
Table 1 Roughness R a and R q (nm) for the studied samples Roughness
Specimen 1 Specimen 2
Step1
Step 2
6. Discussion Step 3
Ra
Rq
Ra
Rq
Ra
Rq
232 142
350 258
151 141
195 256
5 382
6 523
to become more and more rough and the initially rough surface was polished—in three steps. The elastic properties of the GaAs samples are described by the three elastic constants c 11 = 119 GPa, c 12 = 53.4 GPa, c 44 = 59.6 GPa at T = 300 K. All measurements have been realized in the [110] direction, in which the SAW velocity is the highest (V R(0) = 2862 m/s). The central regions of the samples with maximal length 10 mm were used for these measurements. Surface roughnesses R a and R q were measured by means of the Tencor ALPHA STEP 500. Roughnesses were measured in the center of samples over L = 2 mm. These roughness characteristics of the two studied samples at aforementioned three polishing/grinding steps are reported in Table 1.
5. Results The experimental data and fitting curves are plotted in Fig. 3. It is seen that the rough layer causes dispersion of the phase velocity. The rough layer produces loading characteristic—the SAW’s phase velocity decreases with the frequency. On the polished surfaces of the samples the phase velocity variations are less then 1 m/s. Increasing of the surface roughness increases the phase velocity dispersion and vice versa, removal of the rough layer by the stepwise polishing decreases the slope of the dispersion curve. The only fitting parameter is the rough layer effective thickness d. The values of the parameter d are correlated with the corresponding values of the roughness parameters R a and R q. The effective thickness, d, exceeds the measured surface roughness values R a and R q and varies approximately linearly with them. One can hardly expect to have exact coincidence between these quantities, nevertheless, some correlation should exist. The least-squares fitting gives the following relation between the effective thickness, d, and the RMS roughness d,28 þ 1:22Rq :
The damage layer in GaAs wafers have been investigated by the same method in Ref. [21]. The dispersion of SAW had been used for estimations of the thickness of the surface damage layer and its elastic modulus. It was found that the damage layer has considerably lower Young’s modulus than the bulk substrate of GaAs crystal. In general, a damage layer is a medium with physical properties different from those of the bulk crystal. The upper part of the damage layer consists of discontinuous blocks, islands. A transitional region lies under the discontinuous part. It is an elastic medium with an enhanced number of structure defects, such as cracks, pores, dislocations, clusters of vacancies. The defects reduce the Young’s modulus and density of the damage layer. The number of the defects decays with the distance from the surface. Therefore, one should expect that the physical properties are non-homogeneous within the damage layer. The density and, especially, the elastic moduli should vary in wide ranges across the damage layer. The Young’s modulus increases from zero at the surface to the bulk value on the depth of the damage layer. The thicknesses of the discontinuous part and transitional layer depend on the technique used for the slicing of the wafers. To describe the dispersion of the phase velocity, caused by the damage layer one should use some models. In a crude approximation, one can consider the damage layer as a medium without elasticity, having only an effective density (or effective thickness) as an adjustable parameter. For thin layer the value of the dispersion is proportional to the layer thickness. In another model the rough layer is described by some effective elastic constants. The damage layer is considered as a film which thickness and elastic moduli are adjustable parameters. The model looks more exact as compared with the previous simplified description of the roughness, but
ð13Þ
The problem is greatly simplified by the fact, that the rough layer is a small disturbance causing only weak deviations of the SAW phase velocity. Then the dispersion curves have not any peculiarities and can be fitted easily using only one parameter—effective thickness of the rough layer.
Fig. 3. The dependencies of the SAW phase velocity, v R, on the wave frequency, f = x/2p, for different values of the roughness parameter, R q (nm). Symbols denote experimental data. The curves are solutions of the system Eq. (9) for different values of the effective thickness parameter, d (nm), as indicated.
N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
really it results qualitatively in the same phase velocity dispersion curves. The film has also the loading characteristic. The negative slope of the dispersion curve increases with the thickness of the damage layer. There are two different models which give qualitatively the same results. A discontinuous layer with zero elastic modulus and elastic film result in a loading dispersion curve for the phase velocity of SAWs. If one uses the measurements of the SAW dispersion for the estimation of the surface quality, the difference between the models is not very important. But situation is changed when the photoacoustic method is used for the measurements of the elastic moduli of thin films deposited on solid substrates. In the analogous way the surface roughness should influence the dispersion of the Rayleigh mode in the layered structures. The influence of the thin film on the SAW dispersion is small and surface roughness of the film may change substantially the contribution, produced by the film and will influence the values of its elastic moduli measured by the method. Therefore, the investigations of the surfaces and thin films should account for the presence of the surface roughness. One can conclude that, in general, the surface roughness gives rise to the dispersion of the phase velocity of the Rayleigh waves. The effect of the roughness is equivalent to the effect produced by a thin film loading the substrate. The surface roughness can influence substantially the measurements of the elastic moduli of films by the photoacoustic method. The proposed approach is rather simple and can be useful for estimations of the effects produced by surface roughness on the propagation of the SAWs in isotropic and anisotropic substrates and layered structures.
Acknowledgments This work has been supported by Institutional Research Plan No. AV0Z10100522.
189
References [1] D. Schneider, T. Schwarz, Surf. Coat. Technol. 91 (1997) 136. [2] G.W. Farnell, E.L. Adler, W.P. Mason, R.N. Thurston (Eds.), Physical Acoustics, vol. IX, Academic Press, New York, 1972, p. 35. [3] E.I. Urazakov, L.A. Fal’kovskii, Z. Eksp. Teor. Fiz. 63 (1973) 2297; E.I. Urazakov, L.A. Fal’kovskii, Sov. Phys. JETP 36 (1973) 1214. [4] A.A. Maradudin, D.L. Mills, Ann. Phys. (N.Y.) 100 (1976) 262. [5] V.V. Kosachev, Y.N. Lochov, V.N. Chukov, Z. Eksp. Teor. Fiz. 94 (1988) 162; V.V. Kosachev, Y.N. Lochov, V.N. Chukov, Sov. Phys. JETP 67 (1988) 1825. [6] M. de Billy, G. Quentin, E. Baron, J. Appl. Phys. 61 (1987) 2140. [7] D. Leduc, B. Morvan, P. Pareige, J.-L. Izbicki, NDT E Int. 37 (2004) 207. [8] A.A. Tarasenko, L. Jastrabik, N.A. Tarasenko, Eur. Phys. J. Appl. Phys. 24 (2003) 3. [9] O.L. Anderson, W.P. Mason, R.N. Thurston (Eds.), Physical Acoustics, vol. III, part B, Academic Press, New York, 1965, p. 77. [10] D.A. Hugins, W.P. Mason, R.N. Thurston (Eds.), Physical Acoustics, vol. XVIII, Academic Press, New York, 1985, p. 35. [11] B.A. Auld, Acoustics Fields and Waves in Solids, vols. I and II, Krieger Publishing Company, Florida, 1973. [12] H.F. Pollard, Sound Waves in Solids, Applied Physics Series, London, 1977. [13] A. Neubrand, P. Hess, J. Appl. Phys. 71 (1) (1992) 227. [14] J.F. Nye, Physical Properties of Crystals, Oxford University Press, Clarendon, 1957. [15] Lord Rayleigh, Proc. Lond. Math. Soc. 17 (1887) 14. [16] V.V. Kosachev, A.V. Shchegrov, Ann. Phys. (N.Y.) 240 (1995) 225. [17] A.P. Mayer, M. Lehner, Waves Random Media 14 (1994) 321. [18] A.V. Shchegrov, J. Appl. Phys. 78 (1995) 1565. [19] H.F. Tiersten, J. Appl. Phys. 40 (1969) 770. [20] M. Newlands, Philos. Trans. R. Soc. Sci. A245 (1952) 215. [21] D. Schneider, R. Hammer, M. Jurisch, Semicond. Sci. Technol. 14 (1999) 93. [22] D. Schneider, T. Schwarz, B. Schultrich, Thin Solid Films 219 (1992) 92.
Effect of roughness on the phase velocity of Rayleigh waves in GaAs crystals N. Tarasenko, P. Bohac, L. Jastrabik, D. Chvostova, A. Tarasenko * Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 182 21, Prague 8, Czech Republic Received 7 July 2004; received in revised form 25 May 2005; accepted 25 May 2005 Available online 11 July 2005
Abstract The phase velocity dispersion of the Rayleigh waves on surfaces of GaAs crystals having different roughness has been measured by laseracoustic wave equipment. Applied model of a rough film with zero elastic moduli explained the effect of roughness on the acoustic surface wave phase velocity and showed good coincidence with the experimental data. D 2005 Elsevier B.V. All rights reserved. Keywords: Gallium arsenide; Surface roughness; Surface acoustic waves; Phase velocity
1. Introduction A well-known phenomenon of elastic energy propagation along the free surface of a solid of infinite depth is the Rayleigh waves. The waves propagate in directions parallel to the surface only. The amplitude of the waves decays exponentially in direction perpendicular to the surface. Therefore, the wave energy is concentrated near the surface in a narrow layer within a few wavelengths. The waves are localized near the surface of the substrate—the surface acoustic waves (SAWs). Rayleigh waves have attracted considerable theoretical and experimental interest during past decades because of their practical utility in many applications. The SAWs can be also used for investigations of solid surfaces and films, deposited on the surfaces [1]. The damage layers and films adsorbed on the substrate surface cause dependence of the SAW phase velocity on the wave frequency, which is called dispersion. The dispersion depends on the elastic parameters of the layer as well as on the layer density and its thickness. Interpretation of measurements is based on the general theory of elastic waves, which is well developed for the most important cases (see, for example, Ref. [2]). A system * Corresponding author. Tel.: +42 266052165; fax: +42 286581448. E-mail address: [email protected] (A. Tarasenko). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.05.049
of the wave equations for the elastic displacements in the substrate crystal and the layer should be solved and solutions have to satisfy the boundary conditions for the displacements and stresses at the surface and interface between the substrate and layer. It should be noted that the real systems have not such ideal surfaces and interfaces as are supposed in the theory. Typically, the surfaces are far from the perfect flat planes. The surface roughness can be detected in varying degrees on all solid surfaces. The surface roughness leads to the scattering and attenuation of the SAWs. There are some theoretical and experimental investigations of this effect [3 – 6]. It should be noted that the same effect of the interface roughness on Lamb waves propagation have been investigated in Ref. [7]. We have proposed a simple approach for estimation of the phase velocity dispersion caused by the surface roughness in another limiting case when the rough surface is described by an extremely sharp profile function [8]. Using this approach we considered propagation of the surface elastic waves in crystals having cubic symmetry and obtain equations, determining the SAW dispersion for the substrate with rough surface. It was shown that the rough surfaces give rise to the dispersion of the Rayleigh mode. The results are compared with the experimental data of the SAW phase velocity obtained on the GaAs samples. We obtained a
N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
rather good coincidence between the experimental data and calculated curves by fitting only one parameter—the effective thickness of the rough layer proportional to the rough layer density. There is a good correlation between the fitted values of the effective thickness and the measured RMS value of the surface roughness.
2. Model and boundary conditions Let us consider a semi-infinite substrate bounded by a rough surface (see Fig. 1). The substrate fills the half-space x3 fðY r Þ and the rough surface is determined by the profile random function x3 ¼ fðY r Þ. The theory of elastic waves propagation in the solid media and layered systems is well established. We mention here the milestones of the method only, referring the interested readers to the excellent reviews for detailed descriptions [2,9 – 13]. The elastic displacements must satisfy the wave equations in the substrate q
B 2 uj B2 um cijmn ¼ 0; 2 Bt Bxi Bxn
ð1Þ
where u i is the component of the displacement along the coordinate axis x i ; c ijmn are the components of the corresponding elastic stiffness tensor; i, j, m, n = 1, 2, 3. The elastic stiffness tensor c ijmn in Eq. (1) is referred to the coordinate system of Fig. 1 and the summation convention on repeated indices is implied. The solutions of the system are assumed to be in the following form Y uY ðY r ; t Þ ¼ aY expðakx3 Þexp ik Y r vt : ð2Þ They describe plane waves propagating along a direction Y determined by the wave vector k with a phase velocity v. The displacements have constant phase and amplitude along Y any line perpendicular to the vector k . The amplitude of the displacements decays exponentially in the depth of the substrate (the real part of a must be positive only). The rate of decay is chosen as ak in order to simplify formulae.
185
Due to the symmetry properties of the stiffness tensor, its indices are collapsed by pairs as follows: 11Y1, 22Y2, 33Y3, 23, 32Y4, 13, 31Y5, 12, 21Y6 (see, for example, Ref. [14]). In the case of the GaAS substrate, the only nonzero components of its stiffness tensor c ijmn are the following: c 11 = c 22 = c 33, c 12 = c 13 = c 23 and c 44 = c 55 = c 66 m (c 11 c 12) / 2. Then the solution of the system Eq. (1) is reduced to the secular sixth-order algebraic equation in the quantity a. There are six roots a j , j = 1, 2,. . ., 6. Each root a j has a corresponding eigenvector aY ð jÞ. Any partial wave (solution with a definite a j and aY ð jÞ ) satisfies the equations of motion. The general solution will be a linear combination of all partial waves uY ¼
6 X
Y Cj aY ð jÞ exp aj kx3 exp ik Y r vt :
ð3Þ
j¼1
One must choose the non-vanishing values only for C j ’s, which correspond to roots lying within the right half of the complex plane (real part of a j must be positive), to ensure decaying of SAWs with the depth of the substrate, x 3 0. There are only three of them satisfying this condition. In the [110] and [100] directions of the SAW propagation the secular equation splits into second-order and forth-order multipliers. For the [110] direction it has very simple form q þ w a2 a4 2Pa2 þ Q ¼ 0; ð4Þ where: i 1 h wð1 þ rÞ þ rðr þ sÞ=2 ð1 þ sÞ2 ; 2r w Q ¼ ½w þ ðr þ sÞ=2; r P¼
ð5Þ
and other quantities are defined as follows: r = c 11 / c 44, s = c 12 / c 44, q = (r s) / 2 1, w = 1 (v / v t)2, v t = (c 44 / q)1/2 is the transverse shear velocity. One can determine easily the roots and eigenvectors a1;2 ¼ aY ð1;2Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi PF 0
P2 Q;
pffiffiffi 1 ið1 þ sÞa1;2 =p2ffiffiffi ¼ @ ið1 þ sÞa1;2 = 2 A; w þ ðr þ sÞ=2 a21;2
a3 ¼ aY ð3Þ
pffiffiffiffiffiffiffiffiffiffiffiffi q þ w; 0 1 1 B C ¼ @ 1 A: 0 ð6Þ
Fig. 1. Schematic view of the substrate with the rough surface (crosssection in the plane x 2 = 0) and the system of coordinates assumed for the (001)-oriented surface.
The investigations of the propagation of SAWs on the rough surfaces were carried out on the basis of Rayleigh method [15]. In this method the components of the elastic displacement field are expanded in terms of solutions of the equations of motion of the substrate in the region x 3 0. The coefficients in these expansions are determined by continuing the expansions into the rough region 0Vx3 VfðY r Þ, and satisfying the stress-free boundary
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conditions at the surface x3 ¼ fðY r Þ. This leads to the integral eigenvalue equation for the displacement field. The problem is rather complex and only recently a few attempts have been made to investigate rough surfaces of an anisotropic medium [16 – 18]. The Rayleigh approach is valid only when the surface profile is sufficiently smooth. It can not be applied for the profiles consisting of the sharp, abrupt, narrow peaks and valleys as shown in Fig. 1. If the characteristic size of the surfaces inhomogeneities is less than the Rayleigh wavelength k, the SAW can not Fenter_ the rough region. One can imagine the rough surface as an arrangement of microparticles (blocks) of different shape and size, randomly scattered over an ideal flat reference plane, x3 ¼ 0ufmin ðY r Þ, which separates the ideal elastic media, x 3 < 0, and the rough, discontinuous part, 0Vx3 VfðY r Þ. The blocks are independent, there is no lateral interaction between them. Any block moves in accordance with the surface local displacements, but the block displacement do not influence the displacements of the nearest neighboring blocks. It is clear that the elastic waves can not propagate through such Fgranular_, discontinuous medium. Nevertheless, it should influence the propagation of the SAW in the underlying host crystal. The model can be applied to describe the mechanical properties of the island films. Such films are composed of the well-separated particles which do not contact each other. The sizes of these particles vary in wide region depending on the film deposition conditions. Damage layers can be another field of the model application. Surfaces of wafers, sliced by diamond saws from ingots have rather high density of lattice defects, cracks, etc. Such discontinuous, highly porous layers of varying thickness influence also the propagation of SAW. If k exceeds the characteristic size of the blocks, all specific effects related with the shape of the blocks and their geometrical arrangement are smoothed out. As the rough layer cannot support the propagation of the elastic waves one should set the elastic moduli of the medium equal to zero. The only physical property of the layer is the surface density of the blocks, .. It equals to the mass of the blocks covering 1 cm2 of the surface: qˆ .¼ S
Z
fðY r Þdx1 dx2 ;
ð7Þ
S
where S is the area of the sample surface and qˆ denotes the ordinary bulk density of the blocks. The presence of such layer changes the boundary conditions on a flat surface. Instead of the stress-free boundary conditions on the rough surface one has the boundary conditions on a flat surface loaded by the rough layer. To account for the presence of the rough layer we will define the boundary conditions not at the real surface, but at the plane x 3 = 0, which separates the rough layer from the continuous elastic part. The rough layer moves in accord-
ance with the plane displacements. Therefore, the stresses T 3j arising in the plane x 3 = 0 are equal to the inertia forces, due to the acceleration of the rough layer T3j ¼ .
B2 uj at x3 ¼ 0; Bt 2
j ¼ 1; 2; 3:
ð8Þ
It should be noted that these boundary conditions coincide with the ordinary conditions T 3j = 0, if the surface density of the blocks, . = 0, i.e. when the roughness is absent and the outer surface of the film, x 3 = 0, is an ideal flat plane. This approximation reduces the problem of rough surfaces to the simple case of the SAW propagation in the structures, having ideal surfaces, by modification of the boundary conditions. Amplitudes C j ’s must be determined from the boundary conditions Eq. (8), which can be written as follows. Bu1 Bu3 þ c44 ¼ .ðkvÞ2 u1 ; Bx3 Bx1 c44
c11
Bu2 Bu3 þ Bx3 Bx2
¼ .ðkvÞ2 u2 ;
Bu3 Bu1 Bu2 þ c12 þ ¼ .ðkvÞ2 u3 : Bx3 Bx1 Bx2
ð9Þ
In order to have nontrivial solutions, the determinant of the system Eq. (9) must be set to zero: ða hÞað1Þ þ iað1Þ =pffiffi2ffi 1 1 3 ða hÞað1Þ þ iað1Þ =pffiffi2ffi 1 1 3 ðra1 hÞað1Þ þ ispffiffi2ffiað1Þ 3 1
ð2Þ ð2Þ pffiffiffi ða2 hÞa1 þ ia3 = 2 ð2Þ ð2Þ pffiffiffi ða2 hÞa1 þ ia3 = 2 pffiffiffi ð2Þ ð2Þ ðra2 hÞa3 þ is 2a1
a3 h h a3 ¼ 0: 0
The rough layer changes the dispersion equation of SAW. The phase velocity depends on the angular frequency, x = kv. The effect of the rough layer is determined by a small parameter, h = xvd/v2t b 1, where d = ./q is the effective thickness of the damage layer.
3. Quantitative surface evaluations We characterize the surface roughness by ordinary statistic parameters that represented the average properties of the random profiles R a and R q. The R a is the center-line average (CLA) roughness: Z 1 Ra ¼ jfðY r Þ fa jd x1 ; ð10Þ L L and R q is the root mean square (RMS) roughness. It obtained as an average of the squared values of the measured height deviations fðY r Þ fa , then taking the square root of the mean value
N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 2 Rq ¼ ðfðY r Þ fa Þ dx1 : L L
ð11Þ
Here L is the length of the scan and f a is the position of the mean line, which is defined as follows: the total area of the material-filled profile above the mean line should be equal to the total area of voids below the mean line. It follows from the definition, that 1 fa ¼ L
Z
fðY r Þdx1 :
ð12Þ
L
For one-dimensional profiles f a = d(q/qˆ). It coincides with the effective thickness of the rough layer d, if q = qˆ, but may be substantially different in the case of island films. For two-dimensional rough layers the quantities f a and d are different by their definitions. The position of the center-line and the roughness parameters R a and R q depend on the height distribution along some direction, but d depends also on the three-dimensional shapes of the peaks and valleys of the rough layer.
4. Experimental details The non-destructive Laser-Acoustic Wave Analyzer made in the Fraunhofer Institute for Material and Beam Technology (Dresden, Germany) was used for measurements of roughness influence on the SAWs propagation. The analysis is based on measurements of phase velocity of the SAW and yields the dispersion curve. The experimental setup is shown in Fig. 2. The Analyzer is equipped with N 2 laser (wavelength 337 nm). A cylindrical lens focuses the laser beam in a 5 mm abscissa on the
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sample surface. Short laser pulses with energy 400 AJ and duration 0.5 ns generate SAW impulses in wide band spectrum in tens MHz range. The SAW impulses are detected by a piezoelectric transducer using a PVDF foil pressed by a steel wedge to the sample surface. The acquisition of the electrical signals is performed by a digital oscilloscope with sampling rate of 2 GSa/s (bandwidth 500 MHz). The oscilloscope is triggered by photodiode signal generated by the laser pulse. PC performs the signal processing and controls the measuring procedure. The sample and wedge transducer are fixed to a translation stage driven by a computer controlled DC motor with an accuracy Dx = T2 Am. Both move perpendicular to the laser beam to vary the distance between focused laser spot and transducer. The smoothed velocity spectrum is evaluated from one measurement package commonly consist of these 6 stepwise abbreviating distances. The dispersion curve is calculated automatically using the selected range of the velocity spectrum. The software of the Analyzer is based on the solution of the inverse problem of the dispersion of Rayleigh waves on a homogeneous isotropic substrate covered by an isotropic layer [19,20]. Very small variations of the phase velocity can be investigated due to the rather high accuracy of the method. The measuring error of the phase velocity is less than T 1 m/s. The details of the photoacoustic method can be found elsewhere [21,22]. For experimental investigations of the influence of surface roughness on propagation of the Rayleigh mode we have used two wafers sliced from a (001)-oriented GaAs single crystal. The wafer size was <30 2.6 mm. Initially, one of the wafers was rough ground by SiC powder and the latter has polished surface. After each measurement the initially polished surface of the sample was stepwise ground
Fig. 2. Schematic representation of the measuring equipment [21].
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N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
Table 1 Roughness R a and R q (nm) for the studied samples Roughness
Specimen 1 Specimen 2
Step1
Step 2
6. Discussion Step 3
Ra
Rq
Ra
Rq
Ra
Rq
232 142
350 258
151 141
195 256
5 382
6 523
to become more and more rough and the initially rough surface was polished—in three steps. The elastic properties of the GaAs samples are described by the three elastic constants c 11 = 119 GPa, c 12 = 53.4 GPa, c 44 = 59.6 GPa at T = 300 K. All measurements have been realized in the [110] direction, in which the SAW velocity is the highest (V R(0) = 2862 m/s). The central regions of the samples with maximal length 10 mm were used for these measurements. Surface roughnesses R a and R q were measured by means of the Tencor ALPHA STEP 500. Roughnesses were measured in the center of samples over L = 2 mm. These roughness characteristics of the two studied samples at aforementioned three polishing/grinding steps are reported in Table 1.
5. Results The experimental data and fitting curves are plotted in Fig. 3. It is seen that the rough layer causes dispersion of the phase velocity. The rough layer produces loading characteristic—the SAW’s phase velocity decreases with the frequency. On the polished surfaces of the samples the phase velocity variations are less then 1 m/s. Increasing of the surface roughness increases the phase velocity dispersion and vice versa, removal of the rough layer by the stepwise polishing decreases the slope of the dispersion curve. The only fitting parameter is the rough layer effective thickness d. The values of the parameter d are correlated with the corresponding values of the roughness parameters R a and R q. The effective thickness, d, exceeds the measured surface roughness values R a and R q and varies approximately linearly with them. One can hardly expect to have exact coincidence between these quantities, nevertheless, some correlation should exist. The least-squares fitting gives the following relation between the effective thickness, d, and the RMS roughness d,28 þ 1:22Rq :
The damage layer in GaAs wafers have been investigated by the same method in Ref. [21]. The dispersion of SAW had been used for estimations of the thickness of the surface damage layer and its elastic modulus. It was found that the damage layer has considerably lower Young’s modulus than the bulk substrate of GaAs crystal. In general, a damage layer is a medium with physical properties different from those of the bulk crystal. The upper part of the damage layer consists of discontinuous blocks, islands. A transitional region lies under the discontinuous part. It is an elastic medium with an enhanced number of structure defects, such as cracks, pores, dislocations, clusters of vacancies. The defects reduce the Young’s modulus and density of the damage layer. The number of the defects decays with the distance from the surface. Therefore, one should expect that the physical properties are non-homogeneous within the damage layer. The density and, especially, the elastic moduli should vary in wide ranges across the damage layer. The Young’s modulus increases from zero at the surface to the bulk value on the depth of the damage layer. The thicknesses of the discontinuous part and transitional layer depend on the technique used for the slicing of the wafers. To describe the dispersion of the phase velocity, caused by the damage layer one should use some models. In a crude approximation, one can consider the damage layer as a medium without elasticity, having only an effective density (or effective thickness) as an adjustable parameter. For thin layer the value of the dispersion is proportional to the layer thickness. In another model the rough layer is described by some effective elastic constants. The damage layer is considered as a film which thickness and elastic moduli are adjustable parameters. The model looks more exact as compared with the previous simplified description of the roughness, but
ð13Þ
The problem is greatly simplified by the fact, that the rough layer is a small disturbance causing only weak deviations of the SAW phase velocity. Then the dispersion curves have not any peculiarities and can be fitted easily using only one parameter—effective thickness of the rough layer.
Fig. 3. The dependencies of the SAW phase velocity, v R, on the wave frequency, f = x/2p, for different values of the roughness parameter, R q (nm). Symbols denote experimental data. The curves are solutions of the system Eq. (9) for different values of the effective thickness parameter, d (nm), as indicated.
N. Tarasenko et al. / Thin Solid Films 491 (2005) 184 – 189
really it results qualitatively in the same phase velocity dispersion curves. The film has also the loading characteristic. The negative slope of the dispersion curve increases with the thickness of the damage layer. There are two different models which give qualitatively the same results. A discontinuous layer with zero elastic modulus and elastic film result in a loading dispersion curve for the phase velocity of SAWs. If one uses the measurements of the SAW dispersion for the estimation of the surface quality, the difference between the models is not very important. But situation is changed when the photoacoustic method is used for the measurements of the elastic moduli of thin films deposited on solid substrates. In the analogous way the surface roughness should influence the dispersion of the Rayleigh mode in the layered structures. The influence of the thin film on the SAW dispersion is small and surface roughness of the film may change substantially the contribution, produced by the film and will influence the values of its elastic moduli measured by the method. Therefore, the investigations of the surfaces and thin films should account for the presence of the surface roughness. One can conclude that, in general, the surface roughness gives rise to the dispersion of the phase velocity of the Rayleigh waves. The effect of the roughness is equivalent to the effect produced by a thin film loading the substrate. The surface roughness can influence substantially the measurements of the elastic moduli of films by the photoacoustic method. The proposed approach is rather simple and can be useful for estimations of the effects produced by surface roughness on the propagation of the SAWs in isotropic and anisotropic substrates and layered structures.
Acknowledgments This work has been supported by Institutional Research Plan No. AV0Z10100522.
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