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Characterization of interfaces by grazing incidence X-ray scattering
G. Renaud
To cite this version:
G. Renaud. Characterization of interfaces by grazing incidence X-ray scattering. Journal de Physique III, EDP Sciences, 1994, 4 (10), pp.1795-1810. �10.1051/jp3:1994241�. �jpa-00249225�
Classification
Physic-s Abstiacts 61.10F, 68.48, 68.55
Characterization of interfaces by grazing incidence X-ray scattering
G. Renaud
CENG, DRFMC, Service de Physique des Matdriaux et des Microstructures, B-P- 85X,
38041 Grenoble Cedex, France
(Received 28 Februaiy 1994, at.t.epted 20 June 1994)
R4sumd. La gdomdtrie de diffusion des rayons X en incidence rasante et (es expressions de l'intensitd diffractde sont bribvement rappeldes. Les possibilitds de la technique pour la caractdrisation structurale de~ interfaces sont discutdes sur quelques exemples, montrant que des informations compldmentaires telles que la distance interfaciale pour des interfaces cohdrentes, la ddformation, la taille des ilots ou des grains pour des interfaces mcohdrentes, peuvent dtre obtenues par diffdrents balayages dans l'espace rdciproque. Les possibilitds d'analyse de la
structure atomique de phases interfaciale~ sont commentdes.
Abstract. The advantages of the grazing incidence X-ray scattering geometry are emphasized
and the basic expre~sions of scattered intensity are briefly recalled. The application of X-ray
~cattering to the ~tructural characterization of interfaces is di~cussed on selected example~. Well chosen reciprocal space ~can~, both in-plane and out-of plane, are shown to provide complemen- tary information on interracial registry for coherent interfaces ; and on strain deformation, island or grain sizes for incoherent epilayers. The strength of the technique to analyze the atomic ~tructure of
interracial phases is discussed.
1. Introduction.
Grazing incidence X-ray diffraction has been extensively used, since the first study in 1979 ill, to analyze the atomic structure of surface reconstructions of metals, semiconductors and adsorbed monolayers, roughening transitions, surface melting and phase transitions (see
Ref. [2] for recent review~). Although many other techniques are available to investigate the
structure of surfaces, the main advantages of X-rays is that a kinematical approach is sufficient
to analyze the experimental data, and that the reciprocal space resolution, both in position and
amplitude, is much better than with other techniques. However, none of the surface techniques
is available for buried interfaces, whereas the penetration depth of X-rays at glancing angle can
be varied, thus allowing to study interfaces very much the same way as surfaces, non destructively. Most of the structural information on buried interface~ have been obtained by
High Resolution Transmission Electron Microscopy (HRTEM), which gives direct view of the atomic positions at the interface, but only for a very small part, and with limited resolution (~ l.5 Al- The resolution of GIXS is much better, and the information is by nature statistical,
since diffracted intensities are the result of an average over large sample areas.
We begin with a brief presentation of refraction of X-rays under grazing incidence and of the results of diffraction theory applied to crystal surfaces and interfaces. Selected recent
experimental results are then discussed, illustrating different possibilities of the technique.
i~-
f
- 28
/
Fig. I. Grazing incidence X-ray scattering geometry. All notations are defined in the text.
2. Grazing incidence geometry.
The grazing incidence X-ray scattering geometry, shown in figure I, is identical to the three- dimensional case, except that the incident X-ray beam, of wavevector k,, is kept at a glancing
incident angle a, with respect to the surface. The scattered beam, of wavevector
k~, is detected at an angle a~ with respect to the sample surface and at an in-plane angle
2 & with respect to the transmitted beam. The momentum transfer is defined as Q
=
k~ k,, and is often decomposed into a component parallel to the surface Qjj = k~i k,ii, and a
perpendicular component Q~ = k~~ -k,~. The absolute value of Qi is a function of
a, and a~. Qi
=
k (sin a, + sin a~) ; k
=
2 «IA, where A is the wavelength. When
a~ and a~ are very small, Q Qjj, the scattering plane is nearly parallel to the surface, and
diffracting net planes are perpendicular to it. The scattering geometry being defined by the incident beam and detector directions, one has only to rotate the sample about its surface
normal to bring these net planes into diffraction condition, which occurs when they make an
angle & with respect to both the incident and the scattered beam. In this way, the long-range periodicity parallel to the surface is probed. It is often useful to measure the scattered intensity
as a function of Qi, which is in most cases achieved by keeping glancing incidence, but
increasing a~.
3. Refraction of X-rays ; scattering depth.
Because the incident angle is small, it is necessary to consider the effects of refraction at the surface [3]. The refractive index, n, of matter for X-rays is slightly less than unity :
n = ip (1)
=
~~ ~
p ~ (2)
2 " nic~ A P
=
£ (3)
where p is the density, Z the atomic charge, A the atomic weight and p the photoelectric absorption coefficient. As a consequence, the incident beam is refracted, and the transmitted
beam rotates towards the surface. When a, is smaller than a critical value a~, the beam is
totally externally reflected, and only an evanescent wave, which decays over tens of angstroms, is present below the surface. When a, is larger than the critical angle for total
external reflection, the transmitted wave propagates. Typical orders of magnitude are
10~ ~ and p 10~ ~, so that a~
= (2 )"~ 0.2° to 0.5°. Due to time microreversibility,
identical refractive effects occur as a function of the exit angle aj. The perpendicular components of the incident and emergent wave vectors are modified upon crossing the surface
and become complex due to refraction and absorption :
~~.t L ~
~£
, ~°S~
"
>,t 14)
and hence the momentum transfer inside the sample Q[
= k(~ t,'~ becomes complex. The
scattering depth
.I
= I/im (Q[1 (5j
is thus strongly affected by refraction when a, or aj are close to a~. Figure 2 shows the
variation of the scattering depth as a function of a,la~ and ajla~. When aj« a~ and
a~ ma
~,
the scattering depth is of the order of tens of angstroms it increases rapidly to
~~3
ap10a~
- z
°~ lo
'S a=a
~a=0.5c~
ioi f
~ ~qla
Fig. 2. Variation of the scattering depth a~ a function of incident angle, for three exit angles, equal to
half, one time and ten time~ the critical angle, for a Pt surface and 1.51 wavelength.
thousands of angstroms when a~ and a~ are larger than a~, through a transition region where A l 00 h. The incident and exit angles thus allow control of the depth contributing to a given
measurement, which can be varied from lo to 000 h.
4. Scattered intensity.
We will recall briefly the basic expression of the intensity scattered by a crystal. either three- dimensional, quasi-two dimensional, or truncated by a surface. The reader is referred to standard textbooks [4] for a more comprehensive introduction.
4.I THREE-DIMENSIONAL CRYSTAL. Since the interaction of hard X-rays with matter is
small, the kinematical approximation of single scattering is valid in most cases, except for perfect crystals near Bragg scattering. It is also valid under glancing angles [5]. The intensity
scattered by a block-shaped crystal with NJ, N~ and N~ unit cells along the three crystal axes
defined by the vectors aj, a~ and a~ takes the form : I(Q)
~ A (F(Q)(~ (SN~(Q'aj)(~ (SN,(Q a2)(~ (SN~(Q'a3)(~ (6)
where A is a constant and
N-I
S~ (Q a~ = z exp (iQ a~ x n
,
j = 1, 2, 3 (7)
,,=n
F(Ql is the structure factor, which is expressed as a function of all atomic positions
q within the unit cell as
F (QI " I fj exp(iQ r~ (8)
1un,> «11
where f~ it the scattering factor of atom j.
~
Q, a~
sin N
[Sll(Q a~Ii~ =
~
,
j = 1, 2, 3 (9)
~
Q a~
~'~' 2
is the interference function of N diffracting units. In the limit of large N, this function tends to a
periodic array of Dirac delta functions with Q spacing of 2 «la, I.e. the intensity is non-zero
only if Q, aj
= 2 grh, Q, a~
= 2 grk and Q ai
=
2 grf, with h, k, f integers, or, in other
words, if Q is a vector of the reciprocal lattice of basic vectors hi, b~ and b~.
Q
= hbj + kb~ + fb~. When this La0e condition is fulfilled, the intensity is given by I~,t
= A [F~,t[~N( N( N(. (10)
The structure factor now takes the form
~hif
" l~cll ~ e~p ~ ~~ ~'& ~ kYJ ~ f~j)I ~Xp M
J
(j
the summation extends over all atoms of the unit cell f~, x~, y~, z~, M~ are respectively the
scattering factor, fractional coordinates in the unit cell and Debye-Waller factor of atom j.
Qi(°~ l)
2D ROD
Bulk
Bragg peak
Qii (or h)
Fig. 3. Schematic representation of the reciprocal space of a quasi-two dimensional crystal giving rise to continuous rods of diffraction (shaded rods), and of a three-dimensional crystal truncated by a surface, giving rise to the crystal truncation rods (black filled) with maxima at the bulk Bragg positions and
quickly varying intensity in between.
4.2 QUASI-TWO DIMENSIONAL CRYSTAL. Consider now a quasi-two dimensional crystal of thickness a~, taken as the basic cell vector perpendicular to the surface normal. This crystal is handled by setting N~
=
in equation (6). The diffraction is then still sharply peaked in both directions parallel to the surface, but the Laoe condition on Q~ is relaxed, and the intensity is continuous in the out-of-plane direction : the reciprocal space is made of rods perpendicular to the surface plane (Fig. 3). If we still define f by Q ai = 2 grf, f is
now taken as a continuous variable since intensity is present for non integer values of f. The intensity is now given by :
I(f(f)
= A [Fj,,(f)[~N(N(. (12)
The intensity variation along the rod (I.e. as a function of Qi or ii is solely contained in the structure factor ; it is thus related to the z-coordinates of the atoms within the unit-cell of this
quasi-two dimensional crystal. In general, the rod modulation period gives the thickness of the distorted layer and the modulation amplitude is related to the magnitude of the normal atomic displacements.
4.3 THREE-DIMENSIONAL CRYSTAL TRUNCATED BY A SURFACE. A crystal truncated by a
sharp surface (or semi-infinite crystal) can be represented by the product of a step function describing the electron density variation as a function of z, the coordinate perpendicular to the surface, and an infinite lattice. The diffraction pattern is then the convolution of the 3D
reciprocal lattice with the Fourier transformation of the step function. An infinity of Fourier components is necessary to built this latter, so that there remain non zero intensity in between
Bragg peaks as a function off the reciprocal space is again made of rods of intensity, called
crystal truncation rods (CTR), extending perpendicular to the surface, and connecting bulk
Bragg peaks [6]. They are schematically represented in figure 3. The intensity variation as a
n
function of Q~ (or ii can be found by replacing Sll~(Q .a~) by z exp(iQ .a~n~) in
iii ~
equation (6), which gives :
~~~~(~)
~
A (F~,(fjj2 ~
2 S'n~ («f ~' ~~ j3)
",,,,"fl, 'I t ~,?~_#~_i_fi~ fi~ fi'
ROUGH SURFACE EXPANDED LAYER
1o3
~
fl
~ 10~
x rn
ll~
10 _.' ,,.~'
qu ,' i
u ..,
=
=,
I' i I
:
= ii-j
.=' _,~
=
0 K 2K 3K 4K
Fig. 4. Intensity variation along a crystal truncation rod profile, as a function of the out-of plane phaseshift Q .a~
=
2 wt. The full curve shows the CTR profile for a perfectly ~harp surface, as
calculated in equation (13) the dotted curve shows the profile for
a rough surface (I.e, exactly covered by
one half monolayer) and the broken curve for a surface with expanded top layer (Reprinted from Ref. [7]).
The intensity variation of1)/~ (f
as a function off is shown in figure 4. Bragg peaks are found for integer values off, but there remain some intensity in between, even when f is a multiple of h31f-integer, that is when successive net planes scatter out-of phase. At these anti-node
positions, 1[/~(f) and Iii (f) have comparable magnitudes, the intensity diffracted by the semi-infinite lattice is of the order of the intensity diffracted by a single monolayer.
4.4 SURFACE ROUGHNESS AND RELAXATION. If the surface is rough on an atomic scale, the
step function has to be replaced by a smoother function which needs less Fourier components
to be built, so that the iniensities of the CTRS between Bragg peaks is smaller than for a
perfectly sharp surface. The sensitivity to roughness is illustrated in figure 4 where a crystal
truncation rod intensities profile is shown as a function off for a perfectly sharp surface and when this surface is covered by exactly half a monolayer. At anti-node position (f
=
1/2 ), the uncovered surface (half monolayer) of the semi-infinite crystal interfers destructively with the top half monolayer, so that the intensity is strongly reduced with respect to the flat surface
case. This has been used to follow layer-by-layer growth during molecular beam epitaxy
(MBE) of Ge on Ge (I II [8]. The intensity measured at an anti-node position as a function of Ge deposition has an oscillating behavior similar to RHEED oscillation the intensity is
maximum when the surface is terminated by a complete monolayer; minimum when
terminated by half a monolayer, with a decreasing amplitude due to progressive surface
roughening as growth proceeds. The period of the oscillation corresponds to the growth of a
complete layer. A clear advantage of X-ray i>eisus electron ~cattering is that these oscillations
can be quantitatively analyzed using simple kinematical theory on the basis of a model for two- dimensional growth [8, 9].
Figure 4 also illustrates the sensitivity of CTR profiles to a relaxed surface : an expanded top layer introduces a variable phase shift as a function off, which results in an asymmetric Bragg peak profile. This asymmetry can be analyzed to deduce the distance between the last layer and the undisturbed surface.
5. Scattering by an interface.
5.I PERFECTLY COHERENT INTERFACE INTERFACIAL DISTANCE. Figure 5 qualitatively
shows the intensity scattered by an epilayer on top of a substrate, as a function of the in-plane and out-of plane momentum transfer. The semi-infinite crystal gives CTRS, and the epilayer gives rods of intensity which may be peaked at different Q~ values if the out-of plane unit
vector of the epilayer differs from the bulk one, and at different Qi values (as in Fig. 5) if the
epilayer is laterally incoherent. The epilayer rod intensity distribution is broad in Qi because of the finite width of the epilayer, but sharply peaked in plane. When the epilayer is laterally
coherent, the rods appear at the same Ql Position as the substrate CTRS. Analysis of the
intensity as a function of Qi (e.g. during scan of Fig. 5) may in that case yield the interfacial distance and the relation (registry) between the epilayer and the Substrate. An example is given
on the Nisi~/Si( II I) interface [10] obtained by Nickel deposition on Si(I II). The silicide
structure being known, the entire problem is reduced to Ihe determination of the interfacial
separation d. Figure 6 shows the measured CTR profile [10] around a bulk Bragg peak for a 7- layer thick Nisi~ film. The intensity is asymmetric due to interference between the substrate
CTR and the broad seven-slit interference function S~(Q a~), with a phase shift Qi d. A fit allowed the determination of d
= (I, lo ± 0.02 al with high accuracy. Similar measurements have been performed to analyze the relaxation of the last monolayer of crystal surfaces -13]
or the structure of interfaces : the a-Si02/Si(I III [14, 15] and the CaSrF2/GaAs(lll)B
interfaces [16]. Interferences between substrate CTRS and reconstruction rods were also used to determine both in-plane and out-of-plane registry of the Ge(I 1)C2 x 8 reconstruction [17].
Q~(Or 1) ~#~~
peak Substrate
C7R
- E~flayer
Scan 3 ragg
peak
~i Epflyer
ro
z
qjj (or h)
Fig. 5. Schematic representation of the reciprocal space of a substrate with an incoherent epilayer.
Vertical dashed lines are the epilayer rods, which have maxima represented by ellipses substrate CTRS
are also represented. Scan1, 2 and 3 are referred to in the text.
~
w
l
~ l
~j I
« i
« I I
- / ,
~ ' ' '
pj '
z '
uJ '
~ l
z o
o
O-B
OMENTUM RANSFER qao
Fig. 6. -
of the II )/Nisi~ interface (circles), togetherwith the result of the best fit
alculated intensity was btained by introducing a phaseshift between the silicon CTR
structure factor
(dashed line) and the iS12 structure (dashed line). Reprinted from Ref. [10].)
5.2 INCOHERENT INTERFACE MISFIT STRAIN RELAXATION. If the epilayer is not perfectly
lattice-matched to the substrate parallel to the plane of the interface, the rods from the epilayer
will appear at Qj positions different from the substrate CTR, which allows independent
analysis of the epilayer structural properties, such as in-plane and out-of plane relaxation, without interferencei with the substrate. The strain relaxation in the epilayer can be analyzed by performing Qi scans (such as scan 2 of Fig, 5) around in-plane Bragg peaks. The monitoring of the Si(220) and Ge(220) in-plane reflections during the MBE growth of Ge on
Si(001) [18] will be discussed as an example. The misfit, 4 %, between Ge and Si is fairly large so that growth proceeds through a Stranski-Krastanov mode, where islanding occurs after 3-4 « equivalent monolayers » (ML) of uniform deposition. Figure 7 shows a Qi (or h such that Q a
=
2 grh) scan of the (220) silicon reflection for different Ge coverages (in units of
equivalent monolayer, ML, that is l.4 h thickness), taken with a, a~~ l12 a~, which yields a scattering depth A loo I. For ~3 ML, the profile of the Si Bragg peak is
unchanged, which means that the Ge film is perfectly pseudomorphic with the substrate.
Above 3 ML, the base of the Bragg peak becomes asymmetric and a weak shoulder develops.
As growth proceeds, this shoulder moves to lower Q values, until it reaches h =1.92 for
= it ML, which is the expected location of the bulk Ge(220) Bragg peak. Hence, at
=
ML, the Ge film is fully relaxed, while intermediate strain relaxation states are present between 3 ML and II ML. These layers with intermediate in-plane lattice parameter produce
the shoulder at h l.97 between the two peaks at h
=
1.92 and h
=
2, for
= I ML. This
intermediate scattering is not visible on the same scan performed with a, at l/4 a~, which limits the scattering depth to A 40 h, thus confirming that only the last layers
are fully
relaxed. The reader is referred to the original paper [18] for a more complete interpretation of the strain relaxation process through islanding and bending of substrate planes in this system.
