Solving the intensity problem of surface X-ray diffraction using dynamical theory

26 December 1994 PHYSICS LETTERS A

Physics Letters A 196 (1994) 223-228

ELSEVIER

Solving the intensity problem of surface X-ray diffraction using dynamical theory Tsai-Sheng Gau, Shih-Lin Chang Department of Physics, National Tsing Hua University, 30043 Hsinchu, Taiwan, R OC

Received 22 August 1994; revised manuscript received 17 October 1994; accepted for publication 21 October 1994 Communicatedby J. Flouquet

Abstract

We report for the first time a general way of describing the intensity distribution of the surface X-ray diffraction, especially for specular and nonspecular surface normal scans along crystal truncation rods (CTR), using the dynamical theory of X-ray diffraction. New information about the absolute intensity scale, the scattering depth, and the surface roughness is obtained by considering the interaction of X-ray wavefields in crystals. X-ray surface diffraction at grazing incidence is considered as one of the most powerful techniques of probing structures of surfaces and interfaces at atomic scales. The application of this technique for surface/ interface structural studies is currently an actively pursued and interesting research topic in condensed matter physics and surface science. Many important investigations and much progress in this research field have been well documented in the literature. These include studies of the surface roughening transition [ 1 ], the surface order-disorder transition [2 ], the surface melting mechanism [3 ], the incommensurate layer structure [4], the surface and interface morphology [ 5,6 ] and many others [6 ]. Although the usefulness of this diffraction technique has been well proven and widely recognized, there still remain unsolved problems. For example, diffracted intensity data can only be analyzed on a relative basis, because the kinematical theory of Xray diffraction commonly used fails to describe the peak intensity at the Bragg diffraction nodes. Moreover, the scattering depth [ 7 ], a dynamical parameter, has been treated in a kinematical way without

considering the X-ray wavefield interaction under surface Bragg diffraction conditions. Consequently, within this kinematical approach the surface roughness can only be analyzed from the tails of the surface normal scanned profiles [8 ]. The weak intensity at the tails may sometimes cause difficulties in structural analyses. Therefore, a proper way of describing the intensity distribution of surface X-ray diffraction and related physical parameters is needed. In fact there have been attempts to account for the reflectivity and diffracted intensities for the grazing incidence geometry by using the Fresnel formula [ 7 ], the distorted-wave Born approximation [9], and dynamical approaches [ 10,11]. However, a complete description of the intensity distribution, especially for the crystal truncation rods (CTR), is still lacking. Recently, use of Darwin's theory for this same purpose has been reported for a symmetric specular rod [ 12 ], where the scattering vectors have no in-plane components (Fig. 1 ), that is, the rod passes through the origin of the reciprocal lattice. Unfortunately, this approach could not be applied to more general cases in which the asymmetric and large angle off-plane

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224

T.-S. Gau, S. L. ('hang I Phl'sics Letters A 196 (1994) 223-22,';

(a]

,,

(; . D e t e c t o r

/

[]

~

Crvstal Surface ~

-~

(b)

'11

~:!I' ~)~

Fig. I. Schematic of the diffraction geometry in (a) real space, and (b) reciprocal space for a nonspecular rod along L (lhc *3"plane represents the cwstal surface ). For a specular rod, () anti H coincide. diffraction geometry, especially the nonspecular rod where "the in-plane components of the scattering vectors are not null (see Fig. lb), are involved. Besides. in the general situations an analytical expression for the diffracted intensity as well as a numerical calculation cannot be derived or developed in a straightforward manner. Many difficulties are encountered. This fact, from the diffraction and intensity analysis viewpoinL makes the problem of interpreting the intensity of surface X-ray diffraction so challenging and necessaw that a general treatment for this kind of diffraction is desired. In this Letter, we present a general way of describing the intensity distribution for specular and nonspecular CTRs for bare crystal surfaces and for crystals with overlayers. Additional information, nol available from other approaches, about the absolute intensity scale, the (dynamical) scattering depth, and the surface roughness, is obtained. In the following, we first introduce the diffraction geometry and then follow the dynamical theory of Xray diffraction for surface rod scans to derive the fundamental equation of the wavefield and to calculate the wavevectors and the amplitude ratios of the modes of wave-propagation. Finally we d e r b c an expression tor the surface diffracted intensil> by cmploying the proper boundary conditions. Fig. 1 depicts the geometry of the X-ray surface diffraction. For simplicity, we consider only three reciprocal lattice points O, H, and G and the reciprocal rod along L, i.e. HG, of the surface, where O stands also for the origin of the reciprocal lattice. In Fig. l a the incident X-ray beam with the wavcvector k{, making an angle 0~ with the crystal surface is dill

fracted in the ku direction by the H-atomic planes with the reciprocal lattice ~ector OH lying in the plane parallel to the crystal surface and within the crystal. This diffracted beam is then surface specularly reflected along kHs into a detector. The angle between kHs and the surface is 0~. The specularly reflected beam along k<>s of the forward diffi-acted beam ko is not shown in the figure. Inside the crystal several dil;fracted waves with wavevectors Ko(j) and K , (j) are generated (Fig. I b ). The distance between the points (" and C is the excitation error, i.e.. CC'=k<~,. The intensity of the CTR in question is the surface specularly reflected intensit> profile along kHs while the crystal is rotated by varying ~ and O~. (Fig. I a ) or o and 0 (Fig. l b) so that the m o m e n t u m transfer is changing along L flom H to G normal to the surtace (Fig. lb). There the reflection G also comes into pla> in the rod scan. Following thc dvnamical thcor>, derived from Maxwell's equations for X-ray diffraction from complex periodic media, the fundamental equation of Xray wavefields can be written as [ 13,14 ]

!Z,,--2eo)E,,~+Z_.p~I.H~,~-Z

~iP',E~;o+Z
=0,

ZH P21:'o,~+ ( Z ( ) - 2ell )EH,,~+ZJl ~;/:~,~ = 0 ,

=0. /,<, lye/:'(

)~ q- Z{ ;

H E H :r 4- ( Z " - - 2 ~:(; ) E ( ;~ = 0 ,

I }

where E.~, and EH,~arc the a a n d ~r components of the wavefield amplitude E . and the polarization unit vectors 6" and ~ are indicated in Fig. lb. i.e. all the are in the y direction and ~M ×/(M =6"M for M = O , H, and G. The polarization factors, the products of the corresponding two polarization unit vectors, arc [h = c o s o~, p c = c o s 0, and p~=sin oe sin 0. The 2e are defined as 2eM= ( K h - k e ) / k 2 for M = O , H, and G, where k = I/2, 2 being the X-ray wavelength. The electric susceptibility 2'<; is equal to - (e2/mc e ) ( 22/ 7rV) k};, where b'~;, e, m. (', and I'are the structure lhctor, the charge and mass of the electron, the velocity of light, and the crystal unit-celt volume, respec-

T.-S. Gau,S.-L. Chang/ PhysicsLettersA 196 (1994)223-228 tively. The above treatment is commonly used for multi-beam diffraction. The differences between surface diffraction and conventional bulk multiple diffraction are in the form for the 2~ and the difficulties are to find the solutions for the 2~. For the surface case we define the normal component of Ko as Koz = kg related to the excitation error as ko~- kg= k~, ko~ being the z-component ofko. And the 2E can be expressed in terms of 0~, 0f, and kg, keeping the tangential components of the momenta conserved at the crystal surface (Fig. lb), - 2 E o =sin20/-g2,

H, and of the normal components of the B and the D at the crystal boundary (z = 0 ), (eo~-eos,) sin 0i = ~ [kgJKo(j) ]Eoo(j), J

eoa+eo.~= ~/ [Ko(j)/k]EooO'), eo.+eos.= ~ Eo.(j), J

(eo~-eos~) sin 0i= ~gjEo~(J'), J

(3)

for the O-waves,

--2~H =sin20f--g 2 ,

-- 2~c = sin 20f _g2 _ 4 ( L / 2 k - g ) L / 2 k .

225

(2)

For nontrivial solutions of the E the determinant of the coefficients ofEq. ( 1 ) should be null. This sets up the dispersion relation between the wavevectors and the angles 0~ and Of, which is a polynomial equation of degree twelve in g, that is, it is an eigenvalue problem for the nonsymmetric matrix of rank twelve [14]. However, exact solutions with precise accuracy for all the eigenvalues cannot always be found for surface diffraction geometries and the inaccuracy of the determined eigenvalues increases as the Euclidean norm, the square root of the sum of the squares of the elements of the determinant, becomes larger [15 ]. From the diagonal terms of the determinant involving the 2e, the Euclidean norm increases when the incident angle 0~ increases during the surface normal scan with simultaneously varying 0~ and 0r used in a four-circle diffractometer or when a large 0i is used in a normal scan using a z-axis diffractometer. Under these circumstances, the eigenvalue equation cannot be solved in the conventional way but a numerical calculation with an iteration procedure is necessary to retain the solutions asymptotically, for example, the Newton-Ralph method [151. Once the eigenvalues g are determined, the wavevectors and eigenvectors are known and the latter provide the ratios of the wavefield-amplitudes among the diffracted waves. The number o f g is the number of modes of wave propagation. If the crystal is semiinfinite in the z-direction, only six modes are permitted to conserve the total energy. The amplitudes of the E of each mode can be determined from the continuities of the tangential components of the E and

- e H a s i n 0 f = ~ [kgJKn(j)]En~(j), J

e.~= y~ [KH0)/k]EHJj), )

en~ = )-', EH,(j) , J

- e n , sin Of= ~ g/EH,(j),

(4)

J

for the H-waves, and - e c o sin Of= ~ [ k ( g j - L / k ) / K c ( j ) ]EGo(j), J

eoo= ~ [Ko(j)/k]Eo~(j), J

e ~ = Y. E~.(j), J

-e~,sin0f= ~ (gj-L/k)EG,(j), J

(5)

for the G-waves, where e and E are the wavefield amplitudes outside and inside the crystal, respectively. eos is of the specularly reflected wave in the incident direction a n d j indicates the mode. From Eqs. ( 1 ) ( 5 ), eR~, en,, ec~ and eG, can be expressed in terms of the incident wavefield amplitudes eo~ and eo~. Consequently, the intensity for given 0~and Of, namely for a given L, is obtained as

In(Oi, Or)=

(leH,,+eH~ 12+ IeG,~+ eG,~12) sin0f ( [eo¢12+ [eo~ 12) sin 0i (6)

For samples with thin layers on the top, all the twelve modes should be considered in the boundary conditions. The approach can also be applied to specular rods. Although only three reciprocal lattice points are

..'~'~6

T.-S. Gau, S.-L. C h a n g /Physics. Lelter.s .4 196 (1994) ~_.~~ ~-~_, ~ ~'

considered as an example, more lattice points in the vicinity of the rod can be included for a more precise calculation. For the intensity calculation of a whole CTR. the above described procedure has to be repeated tk)r various 0i and 0,. along the CTR. As can be understood, this dynamical approach utilizes the numerical calculation scheme to find the exact eigenvalucs of Eq. ( 1 ) and employ the proper b o u n d a o conditions to calculate the wavefield amplitudes. Hence the interaction among all the wavefields inside the c o s tal is automatically considered. On the contrary, the kinematical theory', based on the Born approximation. which calculates the diffraction intensity as thc superposition of the structure factors of the subdivided layer units, cannot handle the wavefield intcraction and therefore the peak intensity, where the d3namical interaction is appreciably strong. Figs. 2-4 show the calculated intensity distributions o f a NiSi2/Si(l 11 ) specular rod. a (200) n o n specular rod of W (001 ), and a (301) nonspecular rod of A u / G e ( 111 ). respectively, where q is the Miller index referred to the reciprocal lattice of the substrate while (30l) is referred to the conventional surl:ace hexagonal lattice for Ge( 11 I ) surfaces. The e x perimental profiles are taken from Refs. [ 8,16, 17 ] except for the inset of Fig. 2. From these figures well simulated intensity profiles, compared to the experi-

i0 z

>' [.i 10-a aZ

(2 0 0+ill

t,,

,~

Z "

lO-S

0

0.5

!

i .5

2

v! Fig. 3. The nonspccular rod scan of W t 001 ): triangles I e x p c r i mental values [8] with var)ing O, and 0t), solid curve (dynamical ) and dashed curve (kinematical). The origin of the momentum transfer for the dashed curve has been shifted to the right fol comparison.

10 -I [-. 10 -~

(0+q 2+G -2~-q)

m

;z: 10 -s i-

A

10"

&

s 10"

II

0.4

0.8

i.2

11 li)

>, [-,

Fig. 4. The nonspecular rod o l a monolayer of Au on Ge( 11 I ) triangles ( experimental values [ 17 ], with fixed 0i = 1.25 ° ), solid curve (dynamical). and dashed curxe ( k i n e m a t i c a l

~x

1 0 .3

• ~,~B

~t3

Z ka

l

(),995 I.Ol

[.= 10 -6

z: (n n n) I0-"

o

o..l

q

ols

1.2

Fig. 2. The specular rod of 25 ,& NiSi 2 on Si( I 11 ): triangles (ex perimental values [ 16] ), solid curve (dynamical calculation ), dashed curve (kinematical), and dot-dashed curxe (kinemmi cal+Fresnel reflectivily). Inset: the peak intensities of NiSi,,' Si ( 111 ) (A) and NiSi2 ( 111 ) (B) for the 600 ,~ thick NiSi2 sampie (solid curve: dynamical; triangles: measured). The experimental resolution function and crystal mosaicism have been considered for the convoluted intensity curve (solid curve ).

mental ones, are obtained. For NiSi2/Si( 111 ), the reflections (000) and ( 1 l l t of the N iSi2 layer and of the substrate are considered m the calculation. In fact, the eigenvalues of this case can be solved analytically. But for W (001 ), no easy numerical solution can be found and the eigenvalues can only be determined asymptotically, for example, using the Newton-Ralph method. For A u / G e ( I l l ), tile in-plane reflection (02"~) of Au and the off-plane ( 1 3 ] ) of Ge are included in the calculation. For crystals with overlayers, the interference patterns of the rod ( Fig. 2 ) are usually described in the kinematical approach by using a resultant structure factor. To better fit the measured curves, the Fresnel reflectivity has to be added, while in the dynamical

T.-S. Gau, S.-L. Chang / Physics Letters A 196 (1994) 223-228

approach the interference of X-ray wavefields are taken care of by the fundamental equation and boundary conditions such that the Fresnel reflectivity and refraction effect are automatically considered in a very natural way. As shown in Fig. 2, the kinematical calculations with and without Fresnel reflectivity cannot faithfully describe the intensity of the whole rod. In addition, the intensity at the Bragg peak blows up. By contrast, the dynamical calculation provides reasonable values for the intensities at and near the Bragg point. To further verify the peak intensity, a rod for 600/k thick NiSi2 on Si ( 111 ) near the ( 111 ) peak is measured. The calculated intensities of the NiSi2/Si(111) peak and of the incommensurate NiSi2 ( 111 ) agree well with the measured intensities (see the inset of Fig. 2). In Fig. 3, as information about the experimental conditions [ 8 ] is lacking, the dynamical calculated curve has been convoluted with Gaussian distributions of the receiving slits with 0.6 ° opening and of the beam divergence and crystal mosaicism. The latter is assumed as an error function of the scattering depth with a minimum value of 0.15 °. A well fitted curve with a surface roughness parameter (defined in Ref. [8] ) fl=0.25 is obtained (both the 0.6 ° opening and the minimum value 0.15 ° are determined also from the fitting). It should be noted that in order to compare the dynamical and the kinematical calculations with regard to surface roughness we use for simplicity the multiplication of the calculated intensity and surface roughness as the resultant intensity, i.e. I(fl) = I ( Oi, 0f)(l - f l ) 2 / [ 1 + f l 2 - 2flcos( q3a3) ] ,

where q3 and a3 are the momentum transfer and lattice spacing normal to the crystal surface, respectively. In Fig. 4, the dynamical calculation gives a better fit to the measured curve than the kinematical calculation even for the tails. Moreover, the calculation gives the exact peak position of the G reflection shifted from the Bragg node by the amount ( 1 - b ) I X o I/2b sin (20o), where b = sin 0j/sin Ofwith 0o being the Bragg angle of the G reflection ( G = 13i). By knowing the peak position and peak intensity of a CTR, the dynamical calculation therefore provides an absolute scale for the intensity and momentum transfer for rod scans. With this scale, the surface roughness can be determined more accurately from the whole intensity profile measured than

227

from the kinematical approach, because the peak positions and peak widths are not considered in the kinematical fitting. Indeed, from Fig. 3, the best fitted (solid) curve to the measured peak widths and heights gives a surface roughness //=0.25+0.01 ( ~ 1.05/k), instead offl=0.46_+ 0.01 ( ~ 1.98 A) reported in Ref. [ 8 ]. It is also worth noting that in the region far away from the Bragg peaks the dynamical calculation may not be necessarily coincident with the kinematical calculation, especially when more than one crystal boundary are involved. This is mainly due to the kinematical theory not taking into account the boundary. Figs. 2 and 4 exhibit this difference clearly. We estimate also the angular limit over which the use of dynamical theory is a must. For W(001 ) the limit is about 0.02 reciprocal lattice units from the Bragg point for the (202) reflection, which is proportional to lXoX-ol/sin 0o/(sin 0 o - s i n 0i), G being (202). Furthermore, from the excitation errors kgj, the scattering depth, C, can also be determined dynamically as C=Zj (1/kg~)Ex(j), where g) is the imaginary part ofgj and Ex (j) is the modulus of the Poynting vector of mode j normalized to that of the incident beam. Due to the limited space here, detailed results of the scattering depth and surface roughness analysis will be reported elsewhere. In conclusion, we have succeeded in calculating the intensity distribution of a whole CTR using dynamical theory. We have also demonstrated that the dynamical approach presented is quite general and can be applied to specular and nonspecular rods for bare crystal surfaces and overlayers and additional information about the absolute intensity scale and dynamical scattering depth can be deduced from this approach. Also, this calculation procedure provides a new analysis scheme for surface/interface studies using surface X-ray diffraction. The authors are indebted to the National Science Council (NSC) for financial support; SLC thanks K.S. Liang for calling his attention to this intensity problem, H.H. Hung for discussions, and R.G. van Silfhout for providing information. TSG acknowledges the NSC for a graduate fellowship.

228

T.-S. Gau, S.-L. ('hang / Physics Letters A 196 (1994) 223-22~'

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[8] I.K. Robinson, Phys. Rev. B 33 (1986) 3830. [9] G.H. Vineyard, Phys. Rev. B 26 (1982) 4146. [ 10] A.M. Afanas'cv and M.K. Melkonyan. Acta Cryslallogr. A 39 (1983) 207. 1 ] P.L Cowan, Phys. Rev. B 32 I t985) 5437: T. Jach, P.L. Cowan, Q. Shen and M.J. Bedzyk, Phys. Re~. B 39 (1989) 5739: R. Colella, Phys. Rev. B 43 ( 1991 ) 13827: A. Caticha, Phys. Rev. B 47 ( 1993 ) 76. 2] S. Nakatani and T. Takahashi, Surf. Sci. 311 (1994) 433: A. Caticha, Phys. Rev. B 49 (1994) 33. [ 3] S.L. Chang, Multiple diffraction of X-rays in crystals (Springer, Berlin, 1984 ). [ 14 ] R. Colella, Acta Crystallogr, A 30 / 1974 ) 413. [ t 5 ] W . H . Press, S.A. Taukolsky, W.T. Vetterling and B.P. Flannel-y, Numerical recipes in C (Cambridge Univ. Press. Cambridge, 1992 ). [ 16] I.K. Robinson, R.T. Tung and R. Feidenhans'l, Phys. Rev B38 (1988) 3632. [ 17 ] P.B. Howes, C. Norris, M.S. Finney, E. Vlieg and R,G. van Silfhout, Phys. Rev. B 48 (1993) 1632.