Power law behavior in diffraction from fractal surfaces

Transcription

1 Surface Science 409 (1998 L703 L708 Surface Science Letters Power law behavior in iffraction from fractal surfaces Y.-P. Zhao *, C.-F. Cheng 1, G.-C. Wang, T.-M. Lu Department of Physics, Applie Physics an Astronomy, Rensselaer Polytechnic Institute, Troy, NY , USA Receive 2 January 1998; accepte for publication 25 March 1998 Abstract The angular istribution of the iffraction intensity from a fractal surface is proportional to the power spectrum only when V%1, where V is equal to the square of the prouct of the interface with, w, an the momentum transfer perpenicular to the surface, k. However, it is shown that a power law behavior of the form k 2 2a always exists in the large k regime for all values of V, where a is the roughness exponent that is relate to the surface fractal imension D through a=3 D, an k is the momentum s s transfer parallel to the surface. This power law behavior was confirme in a light scattering experiment from a rough Si backsie surface uner ifferent iffraction conitions, i.e. ifferent k values Elsevier Science B.V. All rights reserve. Keywors: Diffraction; Fractal surface; Light scattering; Power law; Power spectrum; Silicon; Surface morphology; Surface roughness Elastic iffraction is a very useful technique for parts: k (=k sin h, the momentum transfer out out probing both the bulk an surface structures of parallel to the sample surface (or momentum materials [1 4]. There are many ifferent iffraction transfer perpenicular to incient momentum irection, an k (=k k cos h, the momentum in out out techniques, such as X-ray iffraction, neutron iffraction, electron iffraction, atom iffraction, transfer perpenicular to the sample surface (or the an light scattering, which allow measurements of momentum transfer parallel to incient momentum ifferent regimes in the reciprocal space. Different irection. The iffraction amplitue can be physical properties of a material can be extracte expresse as a Fourier transform of a function T(r: from the reciprocal space structure. In general, there are two classes of iffraction moes: one is transmis- sion iffraction ( TD, an the other is reflection iffraction (RD. The TD moe is usually use for the stuy of the bulk structure of a material, an U(k= T(re ik r r, (1 P where T(r is, for example, the electron ensity is shown in Fig. 1a. The momentum transfer function for X-ray iffraction, or the transmission k(k=2 k sin 1h ue to the iffraction (or the in 2 out function for light scattering, an r is the threescattere wave vector can be ecompose into two imensional position vector [r=(x,y,z]. The iffraction intensity is then: * Corresponing author. Fax: ( ; zhaoy@rpi.eu 1 On leave from Shanong Normal University, People s I(k=U(kU*(k= D PCPT(rT(r Rr e ik R R. Republic of China. ( /98/$ Elsevier Science B.V. All rights reserve. PII: S ( X

2 L704 Y.-P. Zhao et al. / Surface Science 409 (1998 L703 L708 Fig. 1. Diffraction geometry an reciprocal space geometry for (a transmission iffraction an (b reflection iffraction. The integral in the square brackets is actually the properties, for example, mass fractal an surface autocorrelation function of T(r. Eq. (2 states fractal, the iffraction intensity for a large k will that the iffraction intensity is the Fourier trans- follow a power law in k [5 7]: form of the autocorrelation function of T(r; or, in other wors, the iffraction intensity is proportional I(k3k b for a large k. (3 to the power spectrum of T(r. It is well Here, the fractal is assume to be isotropic. The known that if the material possesses ranom fractal exponent b in Eq. (3 is irectly relate to the

3 Y.-P. Zhao et al. / Surface Science 409 (1998 L703 L708 L705 fractal imension of a material. For the mass fractal with a fractal imension D m, b=d m ; for The iffraction intensity is then: the surface fractal with a fractal imension I(k= PGPeik [h(r+r h(r ] r H eik r r, (7 D, b=6 D [5 7]. s s The situation in RD is slightly more complicate i.e. the iffraction intensity is the Fourier transform compare to TD. The general geometry for RD is of the height ifference function C(k, r= shown in Fig. 1b. The iffraction amplitue for eik[h(r+r h(r ] r, but not the autocorrelation RD can be written as the Fourier transform of a function of the surface height. This is the major surface ensity function D(r: ifference between RD an TD. The function U(k= P D(re ik r r. (4 C(k,r is relate not only to the surface height ifference with a separation r in the x y plane, but also to the iffraction conition efine by k. The function C(k,r is a linear function of In this case, we assume that the surface is homothe autocorrelation function of the surface height geneous, an the surface ensity function can be expresse as a function [4]: only if k is small. Then, the iffuse part of the iffraction intensity is proportional to the power D(r=[z h(r], ( 5 spectrum of surface height [2, 4]. Does the power law relation in Eq. (3 still hol uner any iffracwhere h(r is the surface height at position r[= (x,y] in the average plane of the surface. Then, the iffraction amplitue can be rewritten as: U(k= P e ik h(r e ik r r. (6 Note that there are approximations impose on Eq. (6 if ifferent iffraction techniques are use. For electron iffraction, Eq. (6 is obtaine uner the kinematic iffraction approximation. For other particle iffractions, Eq. (6 can be erive in the first-orer Born approximation. For light scattering, Eq. (6 is the result of the Kirchhoff approximation ( KA. The criteria for the valiity of Eq. (6 for light scattering are that the surface lateral correlation length is much larger than the wavelength of the incient light [8, 9], the root- mean-square slope angle of the surface is very small, an all absolute values of the tangent of the scattere grazing angles are greater than the tangent of the root-mean-square slope angle [9]. It is shown that within the vali range of the KA, the backscattering enhancement is non-existent or very small [8]. Eq. (6 is a two-imensional Fourier transform, whereas Eq. (1 is a three-imensional Fourier transform. The transfer function in Eq. ( 6 is a function of phase, etermine by both the surface height istribution an the iffraction conition. tion conition, incluing a large k, for a fractal surface morphology? In the following, we shall aress this issue quantitatively. Recently, a consierable avancement in the unerstaning of the nature of rough surfaces has been mae base on the concept of self-affinity [10,11]. A self-affine surface is a class of fractal objects that can be escribe by a roughness exponent, which is relate to the fractal imension of the surface. The surface can be escribe using a height height correlation function efine as [10,11]: H(r=[h(r h(0]2=2w2 f Ar jb. (8 Here, w=[h(r h:]21/2 is calle the interface with, an h: is the average surface height. The function f(x is a scaling func- tion, having the following properties: f (x= Gx2a for x%1 1 for x&1. (9 The exponent a is calle the roughness exponent (0 a 1, which escribes how wiggly the surface is. The roughness exponent is irectly relate to the surface fractal imension D by a=+1 D, s s

4 L706 Y.-P. Zhao et al. / Surface Science 409 (1998 L703 L708 where +1 is the imension of the embee where f (x#2ax2a 1. Therefore: space. The lateral correlation length, j, is the 22aC(1+a istance within which the surface heights of any P(k#8paw2k 2 two points are correlate. These three parameters, (jk2ac(1 a w, a, an j, are inepenent from each other, an vary accoring to the processes by which the =4pw2Aj 2ak 2 2a for kj&1, (15 surface morphology is forme. The parameters w, where: j, an a characterize the major statistical properties of a self-affine surface. A= 22a+1aC(a+1. The characteristics of a self-affine surface can C(1 a also be efine through the surface power Therefore, the efinitions in Eqs. (9 an (11 for spectrum, which gives: a self-affine surface are equivalent. ( These relations P(k=w2g(jk, (10 are equivalent only for values of a that are not too close to 1. For a~1, a iscrepancy may occur where the function g(x has the properties: between these two efinitions of a; see Ref. [12]. 1 for k%1 The iffraction profile of a surface is etermine g(x= (11 solely by the function C(k,r. For a ranom an Gk 2a for k&1. fractal surface, C(k,r is the characteristic function of the height ifference separate by r an shoul Note that both scaling functions in Eqs. (9 an contain all of the statistical parameters that escribe (11 give only the asymptotic behaviors of the the surface. For example, for a Gaussian height characteristic functions; the exact forms of these istribute an isotropic surface, Eq. (7 becomes: characteristic functions may vary. For 0 a<1, we can show that Eqs. (9 an (11 are equivalent, in the following. I(k= P e 1/2k2 H(r eik r r. (16 The relation between the surface power spectrum an height height correlation function can be writprofile For a self-affine rough surface, the iffraction ten as: contains a sharp peak resulting from the P(k= P [2w2 H(r]e ik r r. (12 features of the surface on the long-range scale, an a broa iffuse region reflecting the roughness on the short-range scale. The iffraction structure For an isotropic surface: factor can be written as [4]: P(k=2p P [2w2 H(r]rJ0 (krr 02 I(k=(2p2e k2 w2 (k +S (k,k, (17 iff where: =4pw2j2 P 02 [1 f(x]xj0 (kjxx, (13 S iff (k,k = P r[e 1/2k2 H(r e k2 w2 ]eik r, (18 where 1 f(x=1 x2a for x%1, an 1 f(x=0 for x2, an J (x is the zeroth-orer Bessel for a self-affine surface with a Gaussian height 0 function. Integrating Eq. (13 by parts, one has: istribution. Therefore, the power law behavior, if it exists, shoul be exhibite in Eq. (18. We can P(k=4pw2k 2 P jkxj1 (jkxf (x x. (14 express Eq. (18 as: 02 S (k,k =2pj2e V Here, J (x is the first-orer Bessel function. For iff 1 kj&1, the ominant contribution in the integral 2 1 m=1 m! Vm 2 P x[1 f(x]mj0 (k jxx, (19 of Eq. ( 14 comes from the region of small x (%1, 0

5 Y.-P. Zhao et al. / Surface Science 409 (1998 L703 L708 L707 where V=k2 w2, by expaning the integral kernel in a Taylor series. For a small V, only m=1 gives the major contribution to the iffraction intensity, an then S iff (k,k is proportional to the power spectrum shown in Eq. (13. For any V, an for a large k (&j 1, the main contribution to the integration in Eq. (19 is from the region x%1, an therefore, the asymptotic behavior of the integral for a large k can be written as: P 0 2 x[1 f(x]mj0 (k jx x # P 02 x[1 x2a]mj0 (k jxx # P x[1 mx2a]j0 (k jx x. (20 02 Then, base on the result obtaine from Eqs. (14, (15 an (20, one can reuce Eq. (19 to: 2 m S (k,k #2pj 2ae VA iff m=1 m! Vmk 2 2a =2pj 2aVAk 2 2a. (21 For any value of V, the shape of the iffuse profile varies as a power law in k for a large k, as shown Fig. 2. Height height correlation function H(r for a Si backsie in the asymptotic form of Eq. (21. This power sample obtaine from AFM images (100 mm 100 mm. The law relation also hols for surfaces with non- H(r is average over 10 images. Gaussian height istributions [ 13]. The general relationship for a surface embee in +1 imensions can be expresse as: use in the light scattering experiment. The etails will be publishe elsewhere [14]. We change the S (k,k 3k 2a for k &j 1. (22 iff light scattering conitions by varying the incient The FWHM of the iffuse profile is proportional angle h as enote in Fig. 1b. Fig. 3 shows a in to k1/a [4]. Therefore, as k increases, the corre- log log plot of the normalize light scattering sponing power law tail of the iffuse profile intensity profiles for h=84, 80, 76, 70, 66, an woul start at a larger k value. 60, which correspon to V=0.42, 1.38, 1.92, 2.72, To emonstrate our preiction ( Eq. (22, we 3.23, an 3.97, respectively. Despite the ifferences performe an experiment of light scattering from observe in the small k region, all six curves show a rough surface (the backsie of a silicon wafer. a similar power law behavior at large k. The The surface of the Si backsie was first charac- slopes for the tails are 2.90, 2.91, 2.51, 2.70, 2.85, terize by atomic force microscopy (AFM, from an 3.00, respectively. The average slope extracte which the real-space height height correlation from the log log plot is 2.81±0.07. Since we function was calculate (an plotte in Fig. 2. use a slit etector, the asymptotic power law From Fig. 2, we extracte w=2400±70 Å, j= becomes 3k 1 2a instea of 3k 2 2a for a large 5.50±0.02 mm, an a=0.91±0.02 [14]. An k [14]. From this, we etermine that a= in-plane configuration with a etector array was 0.91±0.04, which is quite consistent with the value

6 L708 Y.-P. Zhao et al. / Surface Science 409 (1998 L703 L708 power spectrum, but only for V%1. However, at a large k value, the iffuse profile oes obey the power law relation, S (k,k 3k 2a for all V, iff which epens neither on the iffraction conition nor on the surface height istribution. Therefore, from one iffraction profile, one can irectly extract a of a fractal surface at any V, not just V%1. Acknowlegements This work was supporte by NSF. Fig. 3. Log log plot of the normalize iffraction profiles from a Si backsie uner ifferent iffraction conitions. Note that the tails of all profiles fall in a power law behavior. References obtaine from the AFM. Note from Fig. 3 that [1] J.M. Cowley, Diffraction Physics, North-Hollan Publishas k increases, the power law region shifts to a ing, New York, larger k, which is also consistent with our [2] P. Beckmann, A. Spizzichino, The Scattering of Electropreiction. magnetic Waves from Rough Surfaces, Macmillan, New Note that for this sample, the ratio of the lateral York, correlation length j to the He Ne laser wavelength [3] J.C. Stover, Optical Scattering: Measurement an Analy- sis, 2n e., SPIE Optical Engineering Press, Bellingham, l, j/l, was /6328~9.0, an the ratio of the Washington, interface with, w, to the wavelength, l, w/l, was [4] H.-N. Yang, G.-C. Wang, T.-M. Lu, Diffraction from 2400/6328~0.3. The root-mean-square ( RMS Rough Surfaces an Dynamic Growth Fronts, Worl Scientific, slope angle c of the sample was about 2w/j= Singapore, /55 000~3.5. All of the absolute values of [5] H.D. Bale, P.W. Schmit, Phys. Rev. Lett. 53 ( [6] J.E. Martin, A.J. Hun, J. Appl. Crystallogr. 20 ( the tangent of the scattere grazing angles are [7] P.W. Schmit, in: D. Avnir (E., The Fractal Approach greater than the tangent values of the RMS slope to Heterogeneous Chemistry: Surfaces, Collois, Polymers, angle, c, i.e. tan h >tan c. In our experiment, the Wiley, New York, 1989, p. 67. s h ranges from 150 to 174, which correspons to [8] A. Ishimaru, J.S. Chen, J. Acoust. Soc. Am. 88 (1990 s a range of 30 to 6 for the incient angles with [9] E.I. Thorsos, J. Acoust. Soc. Am. 83 ( respect to the surface plane, or a range of 60 to [10] F. Family, T. Vicsek (Es., Dynamics of Fractal Surfaces, 84 for the incient angles with respect to the Worl Scientific, Singapore, surface normal. These numbers are within the [11] A.-L. Barabási, H.E. Stanley, Fractal Concepts in Surface range of valiity of the KA. Therefore, the criteria Growth, Cambrige University Press, New York, for the KA are satisfie [8,9]. [12] H.-N. Yang, T.-M. Lu, Phys. Rev. B 51 ( [13] Y.-P. Zhao, G.-C. Wang, T.-M. Lu, Phys. Rev. B 55 In conclusion, we show that in general, the ( iffuse iffraction profile from a ranom rough [14] Y.-P. Zhao, I. Wu, C.-F. Cheng, U. Block, G.-C. Wang, surface is not always proportional to the surface T.-M. Lu, to be publishe.