DYNAMICAL X-RAY DIFFRACTION FROM CRYSTALS UNDER GRAZING-INCIDENCE CONDITIONS

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DYNAMICAL X-RAY DIFFRACTION FROM CRYSTALS UNDER GRAZING-INCIDENCE

CONDITIONS

H. Hashizume, O. Sakata

To cite this version:

H. Hashizume, O. Sakata. DYNAMICAL X-RAY DIFFRACTION FROM CRYSTALS UNDER

GRAZING-INCIDENCE CONDITIONS. Journal de Physique Colloques, 1989, 50 (C7), pp.C7-225-

C7-229. �10.1051/jphyscol:1989724�. �jpa-00229888�

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COLLOQUE DE PHYSIQUE

Colloque C7, suppl6ment au nO1O, Tome 50, octobre 1989

DYNAMICAL X-RAY DIFFRACTION FROM CRYSTALS UNDER GRAZING-INCIDENCE CONDITIONS

H. HASHIZUME and 0. SAKATA

Research Laboratory of Engineering Materials, Tokyo Institute Technology, Nagatsuta, Midori, Yokohama 227, Japan

RBsumB

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La diffraction des rayons X par les cristaux parfaits en incidence rasante peut Ptre decrite par la thborie dynamique. Pour les angles d'incidence trks faibles, une des ondes de Bloch ou les deux deviennent dvanescentes. I1 existe ggalement un domaine d'angles d'incidence pour lequel l'onde diffractbe externe est bvanescente. Ces propribt6s ont 6t6 confirmgespar l'enregistrement des profils de rgflectivitb d'un

cristal de germanium $ l'aide du rayonnementsynchrotron. De plus, 1'Btude des signaux de fluorescence a mis en Bvidence l'existence d'ondes stationaires dont les plans nodaux et antinodaux sont perpendiculairesiL la surface.

Abstract

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X-ray diffraction in perfect crystals at grazing angles can be described by existing dynamical theories. A t small glancing angles one or both of the two Bloch waves are evanescent. Also there exists a domain of incidence angles for which the external diffracted is evanescent. These properties have been confirmed in reflectivity curve profiles observed from a germanium crystal using synchrotron radiation. Furthermore, fluorescence signals have evidenced the production of standing waves with nodal and antinodal planes normal to the surface. This can find applications in the determination of registry of overlayer atoms parallel to the surface.

1

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INTRODUCTION

A theory of dynamical X-ray diffraction at grazing angles was worked out by Afanas'ev and Melkonyan for Bragg planes normal to a flat surface of a perfect crystal /I/. They calculated reflectivity profiles for the specular and diffracted beams, which showed unusual double steps and sharp peaks. These features arise from unusual properties of X-ray waves inside and out- side the crystal /2/, which can best be seen by referring to a three-dimensional dispersion surface in reciprocal space. Some of the features have been observed by Cowan, Brennan et al.

/3/ and Sakata and Hashizume / 4 / , who achieved a high angular resolution with synchrotron radiation. The latter authors also extended the theory to cover Bragg planes not exactly normal to the crystal surface, and showed that the wave properties are strongly affected by a small off-normal angle of the Bragg planes / 5 / .

2

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PROPERTIES OF X-RAY WAVES IN PERFECT CRYSTALS

Figure 1 shows a geometry where plane-wave X-rays KO are incident on a flat surface of a thick perfect crystal at a grazing angle of

%

and simultaneously satisfy the Bragg condition for a set of net planes nearly perpendicular to the surface. For a small

%

both refracted and diffracted waves are subject to a momentum change due to total external reflection, and a specular wave Ks and a diffracted wave Kh emerge from the surface. We have here two independent incidence angles,

%

and 8 , in contrast to the ordinary geometries. Our geometry features a diffracting vector h making a small angle, p , to the surface (see Fig. 1).

In reciprocal space, the geometry is described by a three-dimensional dispersion surface with two branches. Here real waves are represented by wave points P . (j = 1, 2), Po and Ph at which line L cuts through the dispersion surface and the reflection spheres, respectively. J L is defined from PoO L = KO, and is normal to the boundary surface. Phase matching of the external and internal waves on the surface requires that Po, P and Ph lie on L. Wave points Pj

J .

represent Bloch waves or wave fields built up by the dynamical coupling of the ko and kh ( = ko

+

h) waves. An inspection of the reciprocal-space construction immediately reveals that no real wave points P. and Ph exist for certain locations of line L. In such cases the relevant wave vectors are imaginary and therefore, the associated waves are evanescent. Clearly, the J surface of reflection sphere 0, on which Po lies, is divided into four regions, I to IV, according to the number of real wave points produced (Fig. 2). Note that each of these region represents a certain domain of

(6 ,

^A6 ) , where A f3 = €J

-

^€JB ^with^€JB being the Bragg angle. Wave points PI and P2 are imaginary when Po is located in regions I through I11 and

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989724

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region I, respectively, and thus no propagating wave is produced in the crystal even when A 8

= 0. Similarly, Ph is imaginary for Po located in regions I and I1 so that the external diffracted wave is evanescent. For Po staying in region I, an ideal total external reflection would take place in non-absorbing crystals. For other regions the reflection would not be total, because the incident energy is partly transferred to crystal waves. Region I1 is a special region where a propagating Bloch wave 2 is present in the crystal, but its h component wave is not continued by any external diffracted wave, because kh2 11 is larger than K /2,5/.

We now consider a rocking curve to be measured by scanning

Qb

for a fixed A 8 . In- creasing

%

is represented in Fig. 2 by the projected wave point Po

'

moving along a line normal to 6 from point A through E. While PoJ runs from A to B, we shall have a nearly totally reflected specular beam. On the other hand, the diffracted intensity will remain close to zero until Po' moves up to D. At points B, D and E, there will be anomalies in rocking-curve profiles. Points B and E define the minimum glancing angles,

91c2

^{and $cl}

,

above which propagating Bloch waves 2 and 1 are produced in the crystal, respectively. These angles are termed the critical angles of total reflection for Bloch waves. Clearly, and @c2 depend on

A

8

,

and $c2 = 0 for A 8 ' s greater than a certain positive value which locates Poy at point G (Fig. 2). At such A 8 ' s no Bloch wave 2 is produced for any $O.

Wave properties are more quantitatively known by solving the dispersion equation for the geometry. For small values of

% ,

p and A @ , it is expressed as /4/

where kl means the perpendicular component of k, K is the vacuum wave number, $ h is the take- off angle of the Kh wave given by

and C is the polarization factor.

x ,,

is a complex quantity proportional to the crystal struc- ture factor Fh. The usual dynamical iheory red;ces eq.(l) to the well-known quadratic form using 1 kol

-

KolJ

<

1 kol+ Kol

I

and ko r K. The first part of this approximation is not valid In our case, however, since 1 kol

I ,

I Kol

I

KK. We thus need to solve the quartic dispersion equation (1). The approximation adopted by the usual theory is to neglect the curva- tures of the dispersion surface in the direction normal to the crystal surface. In absorbing crystals and xh

XK

are complex and so also are the four solutions for kol/K in eq.

(1). For sufficiently thick crystals, we need to choose solutions with Im(kol/K)<O. The number of correct solutions depends upon the values of p , A 8 , C o , and is one or two. An alternative criterion for choosing correct solutions is the direction of the Poynting vector.

3

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WAVE PENETRATION DEPTH

Once the wave vectors are determined, the l/e penetration depths for the Bloch waves into the crystal is calculable from Z l/e)j = 1/p = - 1I m ( k O ) . Figure 3 plots Z(l/e)d calculated for a germanium crystai at

A

8 = 0 Jor the 220 refjection with CuKa X-rays. ote that an abrupt change occurs in Z at = Q I , 9, ?s for the two Bloch waves are bounded by the off-Bragg critical angle @c

,

namely $c2 <

@cial .

This relation, which is obvious from the reciprocal-space construction in Fig. 2, is physically explained as follows. Wave field 1 belonging to the inner branch of the dispersion surface consists of an 0 wave and an H wave which are in phase. The standing wave formed by the interference of these waves has anti- nodes on the atomic planes of the diffracting crystal, and thus sees more electrons than ave- rage. Evidently, a higher electron density means a smaller refractive index. The converse is the case with Bloch wave 2 which consists of an 0 wave and an H wave out of phase. Wave field 2 has standing-wave nodes on the atomic planes. The very shallow penetration depths of the evanescent waves at

%

<qjCj amount only to several tens of Angstroms, 'which is two orders of magnitude smaller than the penetration depths achievable in usual Bragg-case diffractions. The different penetration depths for the two Bloch waves can be explained by different distributions of the standing-wave intensities relative to the atomic planes.

It can be seen from eq. (2) that there is a domain of

( % ,

^A⁰⁾^where 0. For such incidence angles the dispersion equation gives two imaginary solutions or a pair of real and imaginary solutions for non-absorbing crystals. The former incidence angles belong to region I of Fig. 2, while the latter angles belong to region 11. The real solution represents Bloch wave 2, which is the only propagating Bloch wave for region 11. The external diffracted wave

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Kh has a large positive Im(KhL) for this region, and hence its intensity has an appreciable value only immediately above the surface. One can define a penetration depth ffr the external evanescent wave by ZSlle)h = -1/4a1m(KhL), which is equal to several tens of Angstroms in usual conditions /2. /.

4

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REFLECTIVITY CURVE PROFILES

The amplitude of the internal and external waves are determined from the boundary conditions.

These demand the parallel components of magnetic vectors H for the external and internal waves to be continuous on the boundary surface, in addition to the continuity of the parallel components of electric vectors E. The first condition is equivalent to the continuity of the normal components of displacement vectors D /6/. For explicit boundary equations, readers are re- quested to see /5/. The boundary equations, together with the amplitude ratio obtained from the dispersion equation (I), determine the wave amplitudes, whence the reflectivities, Rs and Rh, for the external Ks and K, waves are found for given values of

%

and

A $ .

We pointed out in the previous paper the calculated RS(% ) and Rh (qj0 ) profiles show the following features /4/.

=a. For large values of 1 A 8 ( (off-Bragg conditions), the specular profile Rs ( #

o)

tends to a familiar Fresnel curve with a step at critical angle #c

,

and Rh ($0 )rO. As the exact Bragg position (A 6 = 0) is approached, both Rs(#O ) and R

(PO)

change in characteristic manners depending on the sign of p .

=b. Rs(% ) has double steps for certain values of

A

0 , which are located at and %2

.

=c. There is a domain of

(g

A @ ) where q,(#O)rO. This domain corresponds to the earlier- discussed range where #

a. ^<

^0.

=d. The low-angle cusp in Rs(# 0) and the associated bump in Rh(#o ) for the case of p >O.

This arises from local transition of the diffraction geometry from the Laue case to the Bragg case where the internal kh vector is directed upward to the surface.

=e. The reflectivity profiles are sensitively affected by the sign of p as well as its magnitude.

Some of the features have been confirmed in experiments conducted using highly collimated synchrotron X-rays /3,4/. Figure 4 shows reflectivity curve profiles observed from the (220) plane nearly normal to a (111) surface of a germanium crystal. The left panels show Rs(g ) and Rh(% ) observed with the diffracting vector h directed outward, while the right panels show those observed with h directed inward (see the insets). The slightly off-normal (220) planes allowed the two geometries to be studied by rotating the sample by 180' about its surface normal. The data were corrected for the variable cross section of the intercepted beam. The indicated A 0 values are determined by comparing the experimental profiles with the theoretical ones. Double steps are clearly visible in the specular curves Rs(g ) , while the simultaneously measured diffraction curves Rh(g ) show broad peaks. The double steps are a direct manifestation of wave dispersion during the Bragg reflection. The two sample orienta- tions with p = 3.3 and -3.3 mrad give markedly different profiles. Solid lines in each panel of Fig. 4 show calculated reflectivities taking account of the estimated incident-beam profiles, both in the horizontal and vertical directions, as well as the effects of the curved surface figure of the sample. The experimental and theoretical profiles are largely in agree- ment, but obvious discrepancies are found. These can be ascribed to lattice distortions included in shallow surface layers of the germanium sample.

5

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STANDING-WAVE BEHAVIORS

Wave fields in Laue-case crystals have two modulations with extremely different periods /6,7/.

The standing waves are intensity-modulated along the diffracting vector h with a period of 1/( h (

,

namely the d spacing of the Bragg planes. The nodal and antinodal positions for standing waves 1 and 2 are out of phase by a , and do not depend upon the incidence angles.

The second modulation is perpendicular to the surface with a period equal to the Pendellosung distance, which is usually a few tens of microns. This modulation is absent in a thick Bragg- case crystal. Standing waves formed by a thick Bragg-case crystal have been extensively used for localization of surface or interface atoms. The intensity modulation in this case is along a direction making a large angle to the crystal surface. In contrast, the standing waves for grazing-incidence crystals have modulations nearly parallel to the surface. This offers a unique possibility to determine registry of overlayer atoms parallel to the substrate surface by monitoring secondary emissions from target atoms / 2 / . The feasibility of such experiments

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is examined in Fig. 5, which plots field intensities at various interplanar positions on the X- ray entrance surface. The calculation assumes a Ge (111) crystal reflecting MoKa X-rays with the (220) planes at three different %'s close to the off-Bragg critical angle @c, which is equal to 2.5 mrad. In Fig. 5 the atomic positions are represented by z so that two adjacent (220) planes occupy positions z = 0 and 1. The appreciably different field-intensity profiles for different z's promise experimental determination of atom positions at the interface. An apparent obstacle is the variable dependencies of the field profiles on

gb .

This difficulty can be overcome by recording secondary-emission profiles at a few 6's and fitting the data by least squares. The calculations in Fig. 5 suggest that surface-adsorbed atoms can be localized with an accuracy better than several percents of the substrate d spacing, This compares with that achievable in usual standing-wave experiments using Bragg-case diffractions.

Figure 6a plots GeKa fluorescence signals observed by rocking a surface-passivated Ge crystal through the 220 Bragg position for 17.4 keV X-rays at three fixed values of

%

/8/.

The measurements used an energy dispersive high-purity germanium detector positioned close to the sample surface. Clearly, the GeKa profiles confirm the presence of standing waves in the crystal. Most obvious in Fig. 6a is the variation of the GeKa profile from a peaked profile at

6 <

iC into a dipped profile at iO> q5 via a peak-valley profile at

@o

z d C . This can be explained by the variable relative Fntensities of the two standing waves on the fluo- rescing atoms. Figure 6b shows the calculated fluorescence for Ge atoms on the (220) planes.

The calculations largely reproduce the observed profiles but the sharp dip at A 8 = 0 in the profile for

%

212.3 mrad is missing in Fig. 6a.

REFERENCES

/I/ Afanase'v, A. M. and Melkonyan, M. K., Acta Cryst. A. 39 (1983) 207.

/2/ Cowan, P. L., Phys. Rev. B, 32 (1985) 5437.

/3/ Cowan, P. L., Brennan, S., Jach, T., Bedzyk, M. J. and Materlik, G., Phys. Rev. Lett. 57 (1986) 2399.

/4/ Sakata, 0. and Hashizume, H., Jpn. J. Appl. Phys. 27 (1988) L1976.

/5/ Sakata, 0. and Hashizume, H., Report RLEMTIT (Tokyo Inst. Tech.) 12 (1987) 45.

/6/ James, R. W., Solid State Physics, edited by F. Seitz and D. Turnbull, Academic Press, 15 (1963) pp.53-220.

/7/ Annaka, S., J. Phys. Soc. Japan 23 (1967) 372.

/8/ Hashizume, H. and Sakata, O., The third international Conference on Synchrotron-Radiation Instrumentation, Tsukuba; Proceedings to be published in Rev. Sci. Instrum.

Figure caations

Fig. 1. Geometry of grazing-incidence X-ray diffraction in real space (a) and reciprocal space (b). In (b) plane OAB is parallel to the crystal surface. Drawn for a positive tilt angle p of the net plane.

Fig. 2. Four regions I to IV on reflection sphere 0 projected onto a plane parallel to the crystal surface. The intersections of the reflection spheres are represented by straight lines. L is Laue point. A 8 < 0 in the right of L and A 8 > O in the left of L.

Fig. 3. l/e penetration depths of the two Bloch waves in a Ge crystal at exact Bragg position for the 220 reflection with CuKa X-rays. The upper trace is for Bloch wave 2 and the lower trace for Bloch wave 1. Broken line is for off-Bragg conditions.

Fig. 4. Observed (open circles and crosses) and calculated (solid line) reflectivity profiles for a Ge crystal at fixed A e ' s for the 220 reflection with 1.540 A X-rays. Rs : specular beam, Rh : diffracted beam. The calculated curves include the incident-beam profile and the effects of the non-flat crystal surface.

Fig. 5. Field-intensity distributions at various inter-net plane positions on the surface of a Ge (111) crystal for the 220 reflection. The calculation assumes MoKa X-rays with an angular spread equal to one sixth of the Bragg-reflection width.

Fig. 6. Observed (a) and calculated (b) GeKa fluorescence profiles from a surface-polished Ge crystal for the 220 reflection with 17.4 keV synchrotron X-rays. Intensities are normalized to off-Bragg values.

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Figure 1 Figure 2

GLANCING ANGLE 4- (mrad)

Figure 3

GLANCING A N G L E 4, (mrad)

Figure 4

2.0mrad 2.5rnred 3.omrad

4.0

>

5

3.0

5

^2.0

a dl0 G '

0

- 2 - 1 0 1 2 - 2 - 1 0 1 2 - 2 - 1 0 1 2 H -100 0 100

W 100

(a) (b) A8

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INCIDENCE ANGLE 8 ( ~ r a d )

(a) (b)