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Diffusion in multilayers
M. Piecuch
To cite this version:
M. Piecuch. Diffusion in multilayers. Revue de Physique Appliquée, Société française de physique / EDP, 1988, 23 (10), pp.1727-1732. �10.1051/rphysap:0198800230100172700�. �jpa-00246000�
Diffusion in multilayers
M. Piecuch (*)
Laboratoire de physique des solides UA 155 C.N.R.S., Université Nancy I, France (Reçu le 29 septembre 1987, révisé le 4 janvier 1988, accepté le 5 février 1988)
Résumé. 2014 Les expériences classiques de diffusion sont réalisées sur des distances macroscopiques ;
l’utilisation de multicouches permet d’étendre le domaine de ces expériences au nanomètre. Cet article donne les quelques idées de base permettant de comprendre le transport atomique à des distances aussi courtes et dans des systèmes aussi hétérogènes. Les principaux points des méthodes expérimentales utilisées sont également discutés. Enfin on termine en donnant quelques exemples récents de mise en 0153uvre de ces
méthodes.
Abstract. 2014 Diffusion experiments are usually performed at macroscopic length scales, use of multilayers can
lower these scale down to the nanometer range. This paper describes the main idea governing atomic transport
at such short distance and in such inhomogeneous systems. The basic experimental methods involved are also discussed. Some representative recent works are shortly described.
Classification
Physics Abstracts
66.30 - 68.35F
1. Introduction.
Diffusion experiments are usually performed at
different length scales, for instance, usual tracer
method use mechanical cutting of samples and are
limited to few microns, modern method of analysis
like secondary ion mass spectroscopy (SIMS) profil- ing or Rutherford backscattering (RBS) have spatial
resolutions in the hundred Ângstrôm ranges. The values of diffusion coefficient D which can be attained in reasonable time of experiment t are
related to the length scale 1 by : D = l2/t, that is for
1 - 1 03BCm and t - 105 s one has D ~ 10- 17 m2/s and if 1 - 100 A one has D ~ 10- 21 m2/s. Then if one studies diffusion in multilayers of modulation wavelength
A in the 10 A range one is able to measure diffusion coefficients as low as 10-23 m2/s, this is .one of the major interest of diffusion studies in multilayer systems.
The first experiment on diffusion in modulated structures was performed by Dumond and Youtz [1]
(*) Present address : Laboratoire mixte C.N.R.S. St- Gobain (U.M. 37), centre de recherche de Pont-À-Mous-
son, BP 28, 54703 Pont-À-Mousson Cedex, France.
in 1940. The basic ideas are rather simple : if one has
a modulated composition
where q = 2 03C0n/039B and A is the modulation
wavelength, and if c varies with time according to
Fick law :
Then :
And the diffracted intensity I, measured at time t (using X-rays or neutrons) at Bragg angle Ob (the
modulation vector q is then given by q =
4 03C0 sin 03B8b/03BB where À is the wavelength of neutrons
or X-rays), varies as
Where I0 is the intensity measured at t = 0. With
this simple analysis, it was possible to determine
lower diffusivities than by any other methods.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/rphysap:0198800230100172700
1728
Two major effects and some practical difficulties
are forgotten in Dumond and Youtz analysis :
* the first effect is that, in such step diffusion gradient, equation (2) is no longer valid, the free
energy in a point depending not only on composition
but also on composition gradient ;
* the second effect is the presence of strain.
Practical difficulties occur because one can see that
equation (4) is valid in the kinematical approxi-
mation and if c is not the atomic concentration, but
the density of scattering centers.
The first part of this paper will then be devoted to the discussion of diffusion ; in the second part 1 will discuss some problems arising from X-rays or neu-
tron analysis and finally 1 will give few representative experimental results with some emphasis on amorph-
ous and microcrystalline samples and 1 will show that the sensitivity of the method may be useful for
studying various problems. Interested readers can
find a more extensive review paper by Greer and Spaepen [2].
2. Diffusion equation.
The theorical basis of diffusion in modulated struc- tures was established by Cook and Hilliard [3] using
the Cahn-Hilliard [4] formalism of free energy calculations in non homogeneous systems.
Where F is the total free energy of the system, K is the gradient energy coefficient which is related to interfacial energy between components of multilayer system and f (c ) is the free energy of a unit volume of an homogeneous alloy of composition c. The
diffusion equation is then derived as :
Where D = M f " is an effective diffusion coef-
ficient, M is the atomic mobility (positive) and f" = 03B42f/03B4c2 can be positive or negative.
In presence of strain, equation (6) is altered and become
Where D’ = D + 2 M~2 Y, Y is an elastic modulus and -q is the linear expansion per unit composition change (Cook and de Fontaine [5]). One can see that
the equations become complex, but, in case of modulated composition, solution of equations (6) or (7) can be found as :
With
or
Then any experimental method sensitive to
composition variation can follow diffusion in such
structure, in addition measurements at various q give additional information on chemical properties
of alloys (sign of f ", ratio K / f ").
The last problem with diffusion equation comes
from the existence of thermal fluctuations in equilib-
rium alloys. This point has been analysed by Cook [6], recently Stephenson [7] has extended Cook’s results to amorphous phase. The main result of this
theory is the modified time dependence of A(q), equation (8) becomes :
where R (q, t ) = 1 A (q, t)|2 and ROZ(q) is the
Ornstein-Zernicke structure factor which is given by [7] :
One can see that this expression may give rela- tively different results from equation (8) if
q2 ~ - f"/(2 03BA ).
3. Diffraction problems.
X-ray or neutron diffraction on composition modu-
lated systems can be performed at low angles (satellites of (000) Bragg peak) or near any normal
Bragg peak. The second method has several advan- tages : one can use simple kinematical approximation
for calculation of intensity and peak positions, the peak intensities allow determination of strain and then one has an accurate method to evaluate the various parameters entering in Deff . However, in
case of amorphous or microcrystalline samples,
where bulk diffusion coefficients are poorly known
and where the multilayer method has been exten-
sively used, small angle diffraction is the only possible method, thus, in the following, 1 will
concentrate on small angles. X-ray diffraction by
modulated structure has been recently reviewed by
Kadin and Keem [8] and Underwood and Barbee [9] ; the interested reader can find more information in the paper by Nevot in this volume [10]. They have
shown the necessity of using full dynamical theory in analysing experimental results. Two main methods
are useful : (a) various approximations of perfect crystal dynamical theory [8] ; (b) simulation by optical multilayer theory [10]. In case of diffusion
experiments, the first method seems to be more
useful. However, usually, the data are analysed in
kinematical regime.
The essential input of dynamical theory are :
where B (q ) the Fourier coefficient of c, the volume concentration is different from A (q ) as used in equation (1) which is the fourier coefficient of atomic concentration c (see below Eq. (15)), re is the
classical electron radius (re = 2.8 x 10- 15 m) and 0394F1(2) are the real and imaginary part of X-ray
contrast, and :
a is the amplitude absorbed by a unit layer in a
double pass.
Then if the number N of layers is large, two limits
are rather simple :
(i) if |S(q)| > a, the peak reflectivity is given by :
(ii) if |S(q)| ~ 03B1 one has :
In the first case, analysis of diffusion is difficult at the beginning and only at sufficiently long times
when S(q) becomes small the kinematic approxi-
mation can be applied ; in the second case one has immediately I (q, t ) proportional to |B(q, t)|2. Neu-
tron diffraction has also been used recently for this
kind of problem (Janot et al. [11,12]), the data analysis is simpler because neutron absorption is
weak (except in some pathological cases) and dy-
namical theory is rarely useful. The next problem is
that the various diffusion equation are established for the atomic concentration while X-rays or neutron
diffraction are sensitive to volume concentrations, if
the volume ratio VA/VB of the system constituents is not too far from 1, one may linearise cv as a function of c, and cv obeys the same equation as c, in
other case the problem becomes intractable except
at relatively long times where only the first Fourier
coefficient, A or B, is important and we have :
In this kind of problems, synchrotron radiation
and anomalous X-rays diffraction (Simon et al. [13])
can be a very important tool.
4. Some experimental results.
Experimental results were divided in three parts.
The first is devoted to classical diffusion exper- iments. In the second part 1 discuss very recent
achievements in ion beam mixing process and the third part gives an exemple of solid state reactions in
multilayer systems.
4.1 TRUE DIFFUSION PROBLEMS. - As an example
of experimental results, the iron silicon system
seems to be a good choice : silicon in intermetallics has nearby the same volume as iron [14], silicon has
a large range of solubility in bcc iron (ordering
occurs at 10 % but the structure remains bcc), amorphous silicon iron alloys are stable from 20 to 100 % of silicon [15]. Bruson et al. [16] have studied multilayers of amorphous Fe70Si30 alloys and pure silicon. The modulation wavelength A varies from 20
to 40 A and the relative thickness of silicon in iron- silicon alloys was adjusted to assure a mean silicon
content of 50 %. Figure 1 shows first and second
Fig. 1. - X-rays reflectivity (arbitrary units) versus
diffraction angle 03B8 (q = 2 03C0 sin 03B8 / 03BB).
order satellite about the (000) Bragg peak recorded
with Co Ka X-ray radiation. One can see the good signal to noise ratio and some asymmetries in the
first satellite indicating dynamical condition. Figure 2
Fig. 2. - First satellite of (000) Bragg peak for Fe70Si30/Si multilayer for different annealing times.
1730
shows the first Bragg peak at various times and figure 3 shows the decrease of satellite intensities with time, at three wavelengths ; one can see the initial non-linearity due to three possible effects : (i) structural relaxation of the amorphous phase ; (ii) dynamical effects in X-ray diffraction ; (iii) non
Fig. 3. - Time dependence of Log (I/I0) for different
wavelengths.
negligible Orstein-Zernicke structure factor. Fig-
ure 4 shows the more interesting result of the
analysis by Bruson et al., the q dependence of Deff which was interpreted as f " > 0 and K 0, that
is an ordering system (UFesi ( vFeFe + USiSi)/2, if U
is the interatomic potential) ; it is noticeable that iron silicon crystalline alloys are also ordering sys- tems. However, some problems remain open : the system is strongly non linear (039B as determined by the angular position of the Bragg peak varies with time,
for instance, indicating some variation of the mul-
tilayers density) ; the temperature dependence of K / f " is non monotonic, initial curvature in log (I/I0) vs. time curves are not discussed in details
[16].
Fig. 4. - Dependence of effective diffusion coefficient on
modulation wavelength at different temperatures.
Another example is the silicon-germanium system which was studied by Prokes and Spaepen [17] using X-rays and Janot et al. [11, 12] using neutrons ;
severe discrepancies exist between the two groups : the obtained Deff differ by several orders of magni-
tude. This discrepancy was only discussed in terms of
difference in sample preparation [12] and it seems
that some work is needed varying largely modulation wavelength and temperature, this example is, how-
ever, useful to show the difficulties of the method.
Recently Prokes and Spaepen [18] have studied
mean concentration dependence of the diffusion
coefficient in germanium silicon multilayer systems, and concluded that the apparent initial discrepancies
between the neutron and X-rays experiments is essentially a concentration effect.
4.2 ION BEAM MIXING. - The technique of ion
beam mixing of multilayered samples is now cur- rently used [19]. It is a non equilibrium process which acts in the following way : an energetic beam
is sent to the target ; the layer boundaries are
progressively smoothed out by the induced atomic displacements leading to a mixed layer (Fig. 5). The
Fig. 5. - Ion beam irradiation of a bilayer (metal A/metal B), leading to the formation of a mixed layer A-B at the
interface.
ion beam mixing process may be modelled by a quasi-diffusion process, characterised by a diffusion
coefficient D. The rate of the ion beam mixing
process, namely Dt/03A6, t being the time of ir- radiation and W the irradiating fluency, is generally
measured via Rutherford backscattering techniques [20] ; as mentioned in the introduction this technique
is limited to 100 A and thinner layers cannot be analysed. To look at the initial stages of the ion beam mixing process, Traverse et al. [21] have used grazing X-ray reflectometry on a periodic multilayer system during irradiation.
One of their results, obtained on a NiAu mul- tilayer [21], is shown in figure 6. The reflectivity
curves were recorded on the high precision reflec-
tometer of the Institut d’Optique (Orsay). For the virgin NiAu sample, six Bragg peaks are present.
After irradiation with 1017 He/cm2 at an energy of 190 keV at liquid nitrogen temperature, the Bragg peaks are strongly affected. The reflectivity curves
can be simulated by optical multilayer theory [10].
Fig. 6. - Reflectivity curves of NiAu multilayer. Full curve : as prepared ; dotted curve : after irradiation with
10" He/cm2.
Dt products as low as 40 Â2 can be deduced from the simulation [21]. These values are at least twice as
small as the smaller one obtained by Rutherford backscattering. The use of the artificial multilayer technique to measure the effective diffusivity as-
sociated with ion beam mixing experiments have
also been performed by Park et al. [22] on amorph-
ous Si/Ge multilayers.
In the same kind of problem, Mimault et al. [23]
have proposed a different method of working, they
record the Kiessig interference fringes produced by a
few multilayers and study the mixing process by
determination of the thickness and density of the layers.
4.3 SOLID STATE REACTION. - The last work 1 will
mention, is the study of solid state reaction pro- cesses ; since the pioneering paper of Johnson et al.
[24], several groups worked on amorphisation in multilayers ; the diffraction method was used by
Samwer et al. [25], they have used high angle X-ray intensity to show that amorphisation between cobalt
and zirconium multilayers take place by planar growth of the amorphous phase and release of tensile stress in the Zr layer.
5. Conclusions.
One can see in this introductory paper, that diffrac- tion methods are a very powerful tool for studying stability and interface reactions in multilayer sys- tems, however some progress has to be made in the
use of X-ray dynamical scattering theory to gain a
better understanding of the non linear processes
occurring in diffusion reaction. In this kind of
problems use of synchrotron radiation in anomalous mode can be a decisive step.
Acknowledgments.
1 acknowledge A. Traverse and A. Naudon for
helpful discussion and for sending some results prior
to publication.
1732
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