X-RAY STANDING WAVES

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X-RAY STANDING WAVES

A. Authier

To cite this version:

A. Authier. X-RAY STANDING WAVES. Journal de Physique Colloques, 1989, 50 (C7), pp.C7-215-

C7-224. �10.1051/jphyscol:1989723�. �jpa-00229887�

COLLOQUE DE PHYSIQUE

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

X-RAY STANDING WAVES

A. AUTKIER

Laboratoire de Mineralogie-Cristallographie, Universitks P. et M. Curie et Paris VII, associ6 au CNRS, 4 Place Jussieu, F-75252 Paris Cedex 05, France

RCsumC

-

On rappelle le principe de la mCthode de dktermination de la position des atomes aux surfaces et interfaces cristallines B l'aide d'ondes stationnaires de rayons X et on dome une revue bibliographique des principales applications. Les limites thkoriques de la mCthode sont discutkes.

Abstract

-

The principle of the X-Riij- Standing Wave technique for the location of atoms at crystalline surfaces and interfaces is recalled and a bibliographical review of the applications is given. The theoretical limitations of the technique are discussed.

I - INTRODUCTION

Standing waves are formed when an incident and a reflected wave overlap. Their period is the inverse of the difference between the wave vectors of the two waves. If the incident wave is electromagnetic, there is a space modulated photoelectric absorption at the nodes and antinodes. They constitute therefore a probe which excites localized secondary emission of X-ray fluorescence, photoelectrons and Auger electrons. If it is possible to displace the nodal and antinodal planes while knowing their position with regard to the sample surface and to detect and analyse simultaneously the secondary emission, it is therefore also possible to localize atom positions with respect to this surface, whether they are the atoms of the sample itself or adsorbed or imbedded impurities.

Up to now, a Bragg reflection was used to produce the reflected wave but a recent paper(') has shown that total reflection at a grazing incidence can also be used. In the former case, the period of the nodal planes is equal to the diffracting plane spacing and properties of dynamical diffraction theory are used to displace them. In the latter one, it is remarked that the period and the position of the fiinges depend on the angle of incidence and this effect is used to move the nodes. It is therefore sufficient to change the angle of incidence of an incident plane wave at values below the critical angle of reflection to produce a set of standing waves with a tunable period.

This paper will briefly review the standard method using a Bragg reflection. B.W. Batte~man(~.~) was the first to suggest using this effect for the location of atoms within the unit cell and J. Golovchenko et al.(4) showed the feasability of the technique.

This

method developped rapidly in the eighties and more than 130 papers, theoretical and experimental, had already been published when this paper was written. X-ray fluorescence has been used by most groups but photoemission is also widely used (see for instance (5"9)), Auger electrons somewhat less so(''.".

12). Nearly half of the experimental papers cover work done with synchrotron radiation, the advantage of

synchrotron radiation being not only its intensity but also its tunability, that is the possibility of choosing a wavelength close to the absorption edge of the atoms to be studied.

For descriptions of the principle and reviews of the method and its applications, see (I3" %!.

2

-

PRINCIPLE, OF THE METHOD

Let us consider an incident plane wave on a crystal satisfying Bragg's condition for a certain reflection (Fig. 1). In the region where incident and reflected waves overlap they are coupled in a wave field

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989723

D = Do exp

1-

2n i Ko.r]

+

D, exp [- 2n i \.r]

The intensity of the wave field is equal to

where

y

is the phase of the ratio

6

of the reflected to the incident wave

5

=

1

D , P ~ exp (i y ) and h is the difference between the incident and reflected wave vectors:

It is the reciprocal lattice vector associated with the reflection and the product h.r can be written

where N is an integer and

d,

= l/h. As is well known, h.r = constant is the equation of the reflecting planes. Ad represents a certain position in the unit cell, along the normal to the reflecting planes. The expression of the intensity of the wave field can now be written:

It shows that the periodicity of the nodal planes is equal to that of the reflecting planes and that their position relative to the origin of the unit cell is determined by the value of the phase, y , of the reflected wave with respect to the incident wave.

The value of the phase is to be calculated using dynamical theory of d i f £ r a c t i ~ n ( ~ ~ . ~ ~ ~ - ~ ) , even in cases where the diffracted intensity can

be

calculated according to the kinematical theory, for instance in the case of thin crystals(25).

The most widely used geometrical situation is that of the Bragg case represented on figure 1. It can be s h o ~ n " ~ ~ ~ ~ ) that on the small and high angle sides of the reflection curve this phase is equal to cp,+n and to

cp,,

respectively, where

cph

is the phase of the structure factor, even when absorption is taken into account. This means that, on the low angle side, the nodes lie on the planes of maximum electronic density, while it is the antinodes on the high angle side. This position of the nodes and antinodes respective to the lattice planes is at the origin of the well known effect of anomalous transmission, or Borrmann effect, discovered by BorrmantP7)and explained by von Laue When the crystal is rocked trough the Bragg angle two cases are to be distingushed: the ideal, no absorption case, and the real, absorbing crystal case. In the former case the reflected intensity equal to one during the total reflection domain, the phase is constant on the two flanks of the curve and varies from cp,+n to

cph

during the plateau. In the latter case, the variation is continuous. In both cases the position of the nodes remains constant or varies slightly on the left flank of the rocking curve, then shifts by half a diffracting plane spacing, dJ2, and finally remains constant or practiclly so on the right flank of the rocking curve ( ~ i ~ . 2 ) . 1t is this shift of the nodal planes which enables to explore the unit cell and is at the basis of the technique. Roughly speaking, when the antinodes pass through an atom it emits secondary emission whose detection correlated with the measure of the reflected intensity gives the position and nature of the atom. Actually, one compares the measured variation of the yield of the emitted secondary radiation with the calculated variation of the intensity of the wave field for a given value of Add,. By fitting the experimental and theoretical curves, it is possible to determine the real value of Ad/d,for the emitting atom.

Rocking curve widths are of the order of asecond of arc or less. The experimental difficulty of X-ray standing wave measurements is therefore to measure the yield of secondary emission at accurate angular positions within the reflection range. This can be done in the laboratory using a very sophisticated device or, more easily, using synchrotron radiation. For examples of experimental setups, see (""M). In order to avoid the difficulty due to the small rocking curve width, one group has suggested using Bragg angles close to d2(".'" where rocking c w e s are much wider. In that case the crystal perfection is not as important.

AS has been shown, one standing wave experiment enables to locate atom positions along the normal to the diffracting planes.The accuracy is of the order of a few percents. More information can be obtained using several reflections. Using several orders on the same set of diffracting planes enables to obtain several Fourier coefficients of the distribution function of an impurity('7) and to get information as to the mean amplitude of their vibrations(35).

Using different sets of diffraction planes enables to perform a niangulation and to locate the atom in three dimemions(9* s* 37).

If all the emitting atoms occupy the same site, the corresponding signals are identical, but if they occupy random positions, the observed signal is an average over all possible positions. If there is a ceaain fraction of the atoms which occupy the same site while the remaining ones occupy random positions, the observed signal i s the superposition of the two corresponding signals and can be determined from the values of the two fractions.

Conversely, fitting experimental and theoretical curves enables to determine the fraction of atoms at a given site, or coherent fraction(I7. ^38).

AS the Standing Waves are formed wherever the incident and reflected waves overlap, they exist as well inside as above the crystal (Fig. 1). The measured emitted radiation may therefore come either from impurities at the surface or in the bulk, from atoms in an overlayer above the reflecting crystal, or from the atoms of the crystal itself. Fig.

2 shows that the shape of the angle dependence of the signals which would be emitted by gallium and arsenic using a 111 reflection from GaAs would be very different. As the nature of the emitting atom can be identified unambiguously from the energy of the secondary emission, the phase of the reflected wave and the absolute structure of a non-centrosyrnrnetric structure can be determined. This has been done in the case of and of GaAs(26.40,41). In such a situation, the signal has to be integrated over the penetration depth of the wave field inside the crystal. It is well known that this penetration depth is strongly angle dependent and becomes very small within the total reflection domain

-

this is the extinction effect which has to be taken into account when calculating the theoretical yieldc4w. It is found that this extinction effect completely damps the signal and, in order to decrease its influence, Pate1 and Golo~chenko(~)have suggested to look at the fluorescence at a grazing angle to the surface, thus decreasing the depth from which it comes. The depth distribution of impurities can be analyzed from the energy distribution of emitted photoelectrons(42m 45).

The X-ray Standing Wave method is very sensitive: it i s possible to study less than 0.2 monolayer adsorbed on the surface (J.R. Patel, 1989, private communication).

3

-

APPLICATIONS

As the preceding section shows, the big advantage and the specificity of the X-ray Standing Wave method i s that it can determine with a high accuracy the nature, position and degreee of occupancy of atoms as well in the bulk as at the surface of a crystal or at any kind of interface: solid-solid, solid-liquid, solid-gaz, solid-ultra high vacuum.

a)

Bulk

The method has been applied to perfect or nearly perfect crystals of a various degree of complexity: GaAdZ6.q4'), ZnP(46), ruby(47), g ~ r n e t ( ~ ' . ~ ~ ) -

b) C ~ s t a l surfaces and monolavers of adsorbates

Standing waves have been used successfully to study the surface reconstruction of silicon by covering it with a monolayer of germanium(M. "1. The position of the f i t monolayer of adsorbate has been determined in the following systems:

-

bromine chemisorbed on silicon (111)(5%53.")

-

it is noteworthy to remark that the standing wave method has been able to show that the sites occupied by Br are different in a wet system and in ultra high vacuum

-

GaAs on silicon

-

the f i i layer on the silicon is occupied by arsenic, on substitutional sites 5% above the bulk positions

-

Ga on silicon (111)

"-

the gallium occupies three-fold relaxed substitutional sites

-

As

on

silicon ( 1 11)f57J- the arsenic occupies the top of half of the (1 11) double plane expanded outward by

-

5 % from the bulk terminated silicon lattice

- A s on silicon the study supports the model of As dimers on the Si (100) surface

-

Au on silicon (I 11 )(')

-

the gold is here not on top, but embedded in the substrate

-

P b on germanium the melting of the monolayer of lead has been studied

-

thallium electrodeposited on copper the thallium occupies a two-fold position above the copper atoms, at a different distance depending on whether the copper is oxygen free or oxygen contaminated. In the latter case the copper surface is reconstructed with a contraction of 0.3 d;.

c) Distorted surface lavers and overlavers

The standing wave pattern is formed in the bulk in the perfect lattice and the extends above it in strained surface layers or deposited epilayes. Many authors assume that, if the overlayer is not too thick, the nodes and antinodes remain hooked to the bulk, without any change iaperiodicity. The position of impurities inferred from the standing wave signal is therefore referred to theperfect substrate. This is however no more true for a highly deformed or thick layer which is coherently linked to the bulk. With a suitable model for the strained surface layer the modified positions of the nodes can be obtained by solving Takagi-Taupin equations(s* 18.61*62,63.M). It can be shown that it is only for incidences corresponding to the exact Bragg angle for the substrate that the nodes remain hoked to the bulk positions and that for a surface relaxation of 2%, and an interface one lattice spacing thick, the assumption that the nodes remain hooked to the bulk starts to be invalid at the tenth plane in the overlayer above the substrate(@).

Distorted surface lavers

-

measurement of the position of implanted impurities: As in Bi in silicon(65), P in silicon(62).

-

determination of the strain' in a boron doped silicon surface layer")

-

characterization of p-n junction(67) and of a Schottky barrier"*)

-

localizationof indium atoms in a solid solution of InGaxAs,

-1'69)

Overlavers

When there is a thin regular mismatched layer above a substrate, the total yield from atoms in this overlayer is obtained by summing the contributions from each plane. Vlieg et alA70) developped a method to calculate this sum and interpret the experimental results. This sum for Nplanes with spacing d,above a substrate withlattice spacing

d,

is proportional to

with

h.rk =

cMd, +

^{k y}

where

d,

is the distance of the first plane of the overlayer above the top plane of the substrate and

r

^{= (d,}

- d,)/d,.

The sum is shown to be equal to:

I d

² =

I

^{DJ 2}

+ NI 51 +

²

1 51

fc (sin z ~ y l s i n WJ cos(21~ A

+

^I+J)I

where fc is the coherent fraction and A =dJd,

+

(N

-

l)y/2.

It can be seen that the result depends on three unknowns, the coherent fraction, fc, the distance of the fust plane of the overlayer above the top plane of the substrate,

d,

and the lattice spacing in the overlayer, do,. Vlieg et al.OO) used two samples with known values of the number of planes in the sample to fit the experimental results and determine the three unknowns, while Saitoh et al.v1) showed that it is possible to use one sample only but with two reflections.

Using this type of analysis, it has been possible to show that the fust layer of nickel in a nickel silicide overlayer above Si (11 1) is 7-fold ~oordinated(~2 'O. 73) while the coordination is 5-fold in the case of a cobalt silicide o ~ e r l a y e r " ~ . ~ ~ . ~ ~ ) . Similarly, calcium and strontium are 7-fold coordinated in the first plane of an overlayer of CaxSrl-xF, on GaAs (1 1 l)B(71).

d) Synthetic multilavers

Standing waves have also been established indepth-periodic synthetic multilayers consisting of 10 to 200 layer pairs of alternating high and low density materials such as Langmuir-Blodgett films'',

" "

^'I). They have thus been suggested a possible tool for the study of biological membranes(82).

4

-

CONCLUSION

The X-ray Standing Wave technique is avery versatile, sensitive and accurate method for determining the nature and positions of atoms in periodic or nearly periodic structures. It has the unique feature of being able to compare the same surface with liquid, gaz or ultra high vacuum above it and to study solid-solid interfaces. It has been applied essentially to structures on perfect semiconductor crystals; the small width of rocking curves has as consequence that experimental setups are quite sophisticated, even using the synchrotron radiation which is now widely used. At the accuracy of a few percent which the technique permits it is necessary to take into account the strains which may modify wave field positions are. On the other hand, the successful application of the technique to synthetic multilayered mnicrostructures opens hopes to get structural informations on thin films.

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Fig.

1

- Principle of the Standing Wave method. In the volume where incident, Do, and reflected,

D, waves overlap, interferences occur with a spatial periodicity equal to l/h = 4, diffracting

planes spacing.

Fig. 2 - Variation of the intensity of the standing wave field I DI with position Adld,, within the unit

cell and with angle of incidence q. The situation represented corresponds to a 11 1 reflection on

GaAs. . On the low angle side of the rocking curve (q = - 15, dimensionless number), the nodes lie

on themaxima of electronic density. On the high angle side (q = 15),

it

is theantinodes.. In between,

the nodes have been shifted by half the diffracting planes spacing, 412.