Power law behavior in diffraction from fractal surfaces

Surface Science 409 (1998) L703–L708

Surface Science Letters

Power law behavior in diffraction from fractal surfaces Y.-P. Zhao *, C.-F. Cheng 1, G.-C. Wang, T.-M. Lu Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA Received 2 January 1998; accepted for publication 25 March 1998

Abstract The angular distribution of the diffraction intensity from a fractal surface is proportional to the power spectrum only when V%1, where V is equal to the square of the product of the interface width, w, and the momentum transfer perpendicular 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, ) d d where a is the roughness exponent that is related to the surface fractal dimension D through a=3−D , and k is the momentum s s d transfer parallel to the surface. This power law behavior was confirmed in a light scattering experiment from a rough Si backside surface under different diffraction conditions, i.e. different k values. © 1998 Elsevier Science B.V. All rights reserved. ) Keywords: Diffraction; Fractal surface; Light scattering; Power law; Power spectrum; Silicon; Surface morphology; Surface roughness

Elastic diffraction is a very useful technique for probing both the bulk and surface structures of materials [1–4]. There are many different diffraction techniques, such as X-ray diffraction, neutron diffraction, electron diffraction, atom diffraction, and light scattering, which allow measurements of different regimes in the reciprocal space. Different physical properties of a material can be extracted from the reciprocal space structure. In general, there are two classes of diffraction modes: one is transmission diffraction (TD), and the other is reflection diffraction (RD). The TD mode is usually used for the study of the bulk structure of a material, and is shown in Fig. 1a. The momentum transfer k(k=2|k | sin 1h ) due to the diffraction (or the in 2 out scattered wave vector) can be decomposed into two

* Corresponding author. Fax: (+1) 518 276 6680; e-mail: [email protected] 1 On leave from Shandong Normal University, People’s Republic of China. 0039-6028/98/$19.00 © 1998 Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 98 ) 0 02 7 4 -X

parts: k (=k sin h ), the momentum transfer d out out parallel to the sample surface (or momentum transfer perpendicular to incident momentum direction), and k (=k −k cos h ), the momentum ) in out out transfer perpendicular to the sample surface (or the momentum transfer parallel to incident momentum direction). The diffraction amplitude can be expressed as a Fourier transform of a function T(r):

P

U(k)= T(r)e−ik · r dr,

(1)

where T(r) is, for example, the electron density function for X-ray diffraction, or the transmission function for light scattering, and r is the threedimensional position vector [r=(x,y,z)]. The diffraction intensity is then:

PCP

I(k)=U(k)U*(k)=

D

T(r)T(r−R) dr e−ik·R dR. (2)

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Fig. 1. Diffraction geometry and reciprocal space geometry for (a) transmission diffraction and (b) reflection diffraction.

The integral in the square brackets is actually the autocorrelation function of T(r). Eq. (2) states that the diffraction intensity is the Fourier transform of the autocorrelation function of T(r); or, in other words, the diffraction intensity is proportional to the power spectrum of T(r). It is well known that if the material possesses random fractal

properties, for example, mass fractal and surface fractal, the diffraction intensity for a large k will follow a power law in k [5–7]: I(k)3k−b for a large k.

(3)

Here, the fractal is assumed to be isotropic. The exponent b in Eq. (3) is directly related to the

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fractal dimension of a material. For the mass fractal with a fractal dimension D , b=D ; for m m the surface fractal with a fractal dimension D , b=6−D [5–7]. s s The situation in RD is slightly more complicated compared to TD. The general geometry for RD is shown in Fig. 1b. The diffraction amplitude for RD can be written as the Fourier transform of a surface density function D(r):

P

U(k)= D(r)e−ik · r dr.

(4)

In this case, we assume that the surface is homogeneous, and the surface density function can be expressed as a d function [4]: D(r)=d[z−h(r)],

(5)

where h(r) is the surface height at position r[= (x,y)] in the average plane of the surface. Then, the diffraction amplitude can be rewritten as:

P

U(k)= e−ik) h(r) e−ikd · r dr.

(6)

Note that there are approximations imposed on Eq. (6) if different diffraction techniques are used. For electron diffraction, Eq. (6) is obtained under the kinematic diffraction approximation. For other particle diffractions, Eq. (6) can be derived in the first-order Born approximation. For light scattering, Eq. (6) is the result of the Kirchhoff approximation ( KA). The criteria for the validity of Eq. (6) for light scattering are that the surface lateral correlation length is much larger than the wavelength of the incident light [8,9], the rootmean-square slope angle of the surface is very small, and all absolute values of the tangent of the scattered grazing angles are greater than the tangent of the root-mean-square slope angle [9]. It is shown that within the valid range of the KA, the backscattering enhancement is non-existent or very small [8]. Eq. (6) is a two-dimensional Fourier transform, whereas Eq. (1) is a three-dimensional Fourier transform. The transfer function in Eq. (6) is a function of phase, determined by both the surface height distribution and the diffraction condition.

The diffraction intensity is then: I(k)=

PGP

H

eik)[h(r+r∞)−h(r∞)] dr∞ eikd · r dr,

(7)

i.e. the diffraction intensity is the Fourier transform of the height difference function C(k , r)= ) ∆ eik)[h(r+r∞)−h(r∞)] dr∞, but not the autocorrelation function of the surface height. This is the major difference between RD and TD. The function C(k ,r) is related not only to the surface height ) difference with a separation r in the x−y plane, but also to the diffraction condition defined by k . The function C(k ,r) is a linear function of ) ) the autocorrelation function of the surface height only if k is small. Then, the diffuse part of the ) diffraction intensity is proportional to the power spectrum of surface height [2,4]. Does the power law relation in Eq. (3) still hold under any diffraction condition, including a large k , for a fractal ) surface morphology? In the following, we shall address this issue quantitatively. Recently, a considerable advancement in the understanding of the nature of rough surfaces has been made based on the concept of self-affinity [10,11]. A self-affine surface is a class of fractal objects that can be described by a ‘‘roughness exponent’’, which is related to the fractal dimension of the surface. The surface can be described using a height–height correlation function defined as [10,11]: H(r)= [h(r)−h(0)]2=2w2 f

AB r j

.

(8)

Here, w= [h(r)−h: ]21/2 is called the interface width, and h: is the average surface height. The function f(x) is a scaling function, having the following properties: f (x)=

G

x2a

for x%1

1

for x&1.

(9)

The exponent a is called the roughness exponent (0≤a≤1), which describes how wiggly the surface is. The roughness exponent is directly related to the surface fractal dimension D by a=d+1−D , s s

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where d+1 is the dimension of the embedded space. The lateral correlation length, j, is the distance within which the surface heights of any two points are correlated. These three parameters, w, a, and j, are independent from each other, and vary according to the processes by which the surface morphology is formed. The parameters w, j, and a characterize the major statistical properties of a self-affine surface. The characteristics of a self-affine surface can also be defined through the surface power spectrum, which gives: (10)

P(k)=w2g(jk), where the function g(x) has the properties: g(x)=

G

1

for k%1

k−2a−d

for k&1.

(11)

Note that both scaling functions in Eqs. (9) and (11) give only the asymptotic behaviors of the characteristic functions; the exact forms of these characteristic functions may vary. For 0≤a<1, we can show that Eqs. (9) and (11) are equivalent, in the following. The relation between the surface power spectrum and height–height correlation function can be written as:

P

P(k)= [2w2−H(r)]e−ik · r dr.

(12)

For an isotropic surface: P(k)=2p

P

2 0

=4pw2j2

P

[1−f (x)]xJ (kjx) dx, 0

(13)

0 where 1−f(x)=1−x2a for x%1, and 1−f(x)=0 for x2, and J (x) is the zeroth-order Bessel 0 function. Integrating Eq. (13) by parts, one has:

P

2

jkxJ (jkx) f ∞(x) dx. (14) 1 0 Here, J (x) is the first-order Bessel function. For 1 kj&1, the dominant contribution in the integral of Eq. (14) comes from the region of small x (%1), P(k)=4pw2k−2

P(k)#8paw2k−2

22aC(1+a) (jk)2aC(1−a)

=4pw2Aj−2ak−2−2a for kj&1,

(15)

where: A=

22a+1aC(a+1) C(1−a)

.

Therefore, the definitions in Eqs. (9) and (11) for a self-affine surface are equivalent. ( These relations are equivalent only for values of a that are not too close to 1. For a~1, a discrepancy may occur between these two definitions of a; see Ref. [12].) The diffraction profile of a surface is determined solely by the function C(k ,r). For a random and ) fractal surface, C(k ,r) is the characteristic function ) of the height difference separated by r and should contain all of the statistical parameters that describe the surface. For example, for a Gaussian height distributed and isotropic surface, Eq. (7) becomes:

P

I(k)= e−1/2k2)H(r) eikd · r dr.

(16)

For a self-affine rough surface, the diffraction profile contains a sharp d peak resulting from the features of the surface on the long-range scale, and a broad diffuse region reflecting the roughness on the short-range scale. The diffraction structure factor can be written as [4]: I(k)=(2p2)e−k2)w2 d(k )+S (k ,k ), d diff d ) where:

[2w2−H(r)]rJ (kr) dr 0 2

where f ∞(x)#2ax2a−1. Therefore:

(17)

P

S (k ,k )= dr[e−1/2k2)H(r) −e−k2)w2 ]eikd · r, diff d ) (18) for a self-affine surface with a Gaussian height distribution. Therefore, the power law behavior, if it exists, should be exhibited in Eq. (18). We can express Eq. (18) as: S (k ,k )=2pj2e−V diff d ) 2 2 1 ×∑ x[1−f (x)]mJ (k jx) dx, (19) Vm 0 d m! m=1 0

P

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where V=k2 w2, by expanding the integral kernel ) in a Taylor series. For a small V, only m=1 gives the major contribution to the diffraction intensity, and then S (k ,k ) is proportional to the power diff d ) spectrum shown in Eq. (13). For any V, and for a large k (&j−1), the main contribution to the d integration in Eq. (19) is from the region x%1, and therefore, the asymptotic behavior of the integral for a large k can be written as: d 2 x[1−f (x)]mJ (k jx) dx 0 d 0 2 x[1−x2a]mJ (k jx) dx # 0 d 0 2 # x[1−mx2a]J (k jx) dx. 0 d (20) 0 Then, based on the result obtained from Eqs. (14), (15) and (20), one can reduce Eq. (19) to:

P

P P

2 m S (k ,k )#2pj−2ae−VA ∑ Vmk−2−2a diff d ) d m=1 m! =2pj−2aVAk−2−2a . (21) d For any value of V, the shape of the diffuse profile varies as a power law in k for a large k , as shown d d in the asymptotic form of Eq. (21). This power law relation also holds for surfaces with nonGaussian height distributions [13]. The general relationship for a surface embedded in d+1 dimensions can be expressed as: S (k ,k )3k−d−2a for k &j−1. (22) diff d ) d d The FWHM of the diffuse profile is proportional to k1/a [4]. Therefore, as k increases, the corre) ) sponding power law tail of the diffuse profile would start at a larger k value. d To demonstrate our prediction ( Eq. (22)), we performed an experiment of light scattering from a rough surface (the backside of a silicon wafer). The surface of the Si backside was first characterized by atomic force microscopy (AFM ), from which the real-space height–height correlation function was calculated (and plotted in Fig. 2). ˚ , j= From Fig. 2, we extracted w=2400±70 A 5.50±0.02 mm, and a=0.91±0.02 [14]. An in-plane configuration with a detector array was

Fig. 2. Height–height correlation function H(r) for a Si backside sample obtained from AFM images (100 mm×100 mm). The H(r) is averaged over 10 images.

used in the light scattering experiment. The details will be published elsewhere [14]. We changed the light scattering conditions by varying the incident angle h as denoted in Fig. 1b. Fig. 3 shows a in log–log plot of the normalized light scattering intensity profiles for h=84°, 80°, 76°, 70°, 66°, and 60°, which correspond to V=0.42, 1.38, 1.92, 2.72, 3.23, and 3.97, respectively. Despite the differences observed in the small k region, all six curves show d a similar power law behavior at large k . The d slopes for the tails are 2.90, 2.91, 2.51, 2.70, 2.85, and 3.00, respectively. The average slope extracted from the log–log plot is −2.81±0.07. Since we used a slit detector, the asymptotic power law becomes 3k−1−2a instead of 3k−2−2a for a large d d k [14]. From this, we determined that a= d 0.91±0.04, which is quite consistent with the value

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power spectrum, but only for V%1. However, at a large k value, the diffuse profile does obey the d power law relation, S (k ,k )3k−d−2a for all V, diff d ) d which depends neither on the diffraction condition nor on the surface height distribution. Therefore, from one diffraction profile, one can directly extract a of a fractal surface at any V, not just V%1.

Acknowledgements This work was supported by NSF. Fig. 3. Log–log plot of the normalized diffraction profiles from a Si backside under different diffraction conditions. Note that the tails of all profiles fall in a power law behavior.

obtained from the AFM. Note from Fig. 3 that as k increases, the power law region shifts to a ) larger k , which is also consistent with our d prediction. Note that for this sample, the ratio of the lateral correlation length j to the He–Ne laser wavelength l, j/l, was 55 000/6328~9.0, and the ratio of the interface width, w, to the wavelength, l, w/l, was 2400/6328~0.3. The root-mean-square (RMS) slope angle c of the sample was about E2w/j= 2400/55 000~3.5°. All of the absolute values of the tangent of the scattered grazing angles are greater than the tangent values of the RMS slope angle, c, i.e. |tan h |>tan c. In our experiment, the s h ranges from 150° to 174°, which corresponds to s a range of 30° to 6° for the incident angles with respect to the surface plane, or a range of 60° to 84° for the incident angles with respect to the surface normal. These numbers are within the range of validity of the KA. Therefore, the criteria for the KA are satisfied [8,9]. In conclusion, we show that in general, the diffuse diffraction profile from a random rough surface is not always proportional to the surface

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