Iterative solution of the inverse problem of dynamic diffraction in inhomogeneous
S. G. Podorov and V. I. Punegov
Syktyvkar State University, 167001 Syktyvkar, Russia
?Submitted May 28, 1997?
Zh. Tekh. Fiz. 69, 39–42 ?March 1999?
An iterative method for solving the inverse problem of dynamic x-ray diffraction in crystalline
layers that are inhomogeneous across their thickness is developed. The structural
characteristics of an InGaAsSb/AlGaAsSb/GaSb heteroepitaxial system are calculated using the
method. © 1999 American Institute of Physics. ?S1063-7842?99?00703-5?
One of the central problems in solid-state physics is the
investigation of the spatial distributions of structural charac-
teristics of crystals using x-ray diffraction. This problem has
taken on crucial significance in the last 20–25 years in con-
nection with the development of new semiconductor devices.
Despite the large number of publications devoted to this sub-
ject ?see, for example, Refs. 1–10?, a generally accepted,
universal method for solving the inverse problem of x-ray
diffraction has still not been developed. Therefore, the search
continues for new avenues for progress in this area of solid-
In Ref. 8 we developed a method for solving the inverse
problem within the kinematic approximation. Alongside
some other approaches, this method has been used to obtain
the spatial structural characteristics of a thin graded single-
crystal AlGaAs film on a thick GaAs substrate.10However,
this method is not applicable to relatively thick heteroepi-
taxial systems, since the kinematic ?Born? approximation is
valid only for thin layers. Therefore, the purpose of the
present work is to develop a more general approach to the
solution of the inverse problem within an iterative procedure.
2. BASIC EQUATIONS
With no loss of generality, let us consider the symmetric
Bragg diffraction of x rays from a crystalline layer of thick-
ness l, which is inhomogeneous with respect to the depth z
and lies on a thick ideal substrate. Let a monochromatic
x-ray plane wave impinge on the surface of the crystal under
investigation at an angle ?. The diffraction of x rays in a
one-dimensionally distorted crystal is described by the
where E0(z) and Eg(z) are the amplitudes of the transmitted
and reflected waves.
The parameters in ?1? have the following form in the
generally accepted notations:7
Here g is the diffraction vector, and ??????0is the angu-
lar mismatch. The substrate Bragg angle is usually taken as
the reference angle ?0. Distortions of the crystal lattice struc-
ture in the system ?1? are assigned by the function U(z)
Thus, within the problem under consideration, structural
distortions in the test object are specified by the crystal-
lattice strain field ?d(z)/d and by the presence of defects,
whose type, concentration, and dimensions influence the val-
ues of the static Debye–Waller factor
The boundary conditions for the Bragg case are E0(0)
?1 and Eg(l)?Rs(?), where Rs(?) is the amplitude reflec-
tion coefficient of the substrate. Using a known procedure
we can reduce the Takagi equations ?1? to a nonlinear equa-
tion of the Riccati type ?Taupin’s equation?11
with the boundary condition Rg(z,?)?z?L?Rs(?). Equation
?3? underlies the algorithm for solving the problem posed.
Here Rg(z?0,?) is the amplitude reflection coefficient of
the inhomogeneous crystal under investigation.
3. ITERATIVE PROCEDURE FOR SOLVING THE INVERSE
tion coefficient of a kinematic layer of thickness l and is a
solution of Eq. ?3? when the condition ??g?0 is satisfied.
Let Rgbe the reflection coefficient of an inhomogeneous
lei?zU(z) dz, which describes the amplitude reflec-
TECHNICAL PHYSICSVOLUME 44, NUMBER 3MARCH 1999
299 1063-7842/99/44(3)/4/$15.00© 1999 American Institute of Physics
layer in the general case of dynamic diffraction. The proce-
dure for finding the function U(z) will be regarded as the
solution of the nonlinear functional equation
where the asterisk denotes the complex conjugate.
The function U(z) is the unknown parameter in this
equation. Let us define the operator A(U)V:
where a is a parameter.
Equation ?4? can then be written in the following form:
whose solution is also the solution of the inverse problem of
x-ray diffraction in an irregular crystalline layer. The follow-
ing iterative method is proposed for solving the functional
Hence we have
The iteration formula ?7? is the basic relation for solving
the inverse problem of dynamic diffraction in a one-
dimensionally distorted crystal.
4. CALCULATION OF THE STATIC DEBYE–WALLER
FACTOR AND THE STRAIN PROFILE
From the relation ?7? we can find the iterative solution
for the static Debye–Waller factor and the strain profile of
the structure under investigation. To this end, we perform
Fourier transformation in ?7? and express the solution in
terms of Ig(?), Rg?U(n)?, and Rk?U(n)?. For the static
Debye–Waller factor we obtain
The lattice strain profile is found from the iteration for-
where the derivative has the form
The use of a model of a crystal without defects, for
which B(0)?1, as the initial approximation for calculations
based on Eq. ?8? is proposed. Strictly speaking, Eqs. ?8? and
?9? are approximate, since the angular integration interval
?? is bounded. Because of this constraint the Fourier trans-
formation cannot be considered rigorous. This, in turn,
causes the calculated profiles to have an oscillatory charac-
ter. All this affects the convergence of the iterative proce-
dure. Improvement of the convergence is possible if regular-
ization methods are employed.
5. REGULARIZATION PROCEDURES
The iterative algorithm can be regularized using the con-
volution of the solutions sought for ?8? and ?9? with Gauss’s
The values of the coefficients ? in ?11? and ?12? were
adjusted so as to smooth the oscillatory behavior of B(z) and
f(z) due to the constraints on the angular integration interval
in ?7?–?10?. We note that in calculating the angular distribu-
tion of the diffraction intensity the ? and ? polarizations of
the waves were taken into account and that the convolution
with the instrumental function of the monochromator was
6. DETERMINATION OF THE STRUCTURAL
CHARACTERISTICS OF AN INHOMOGENEOUS
The computational iterative procedure developed is ap-
plicable to the structural diagnostics of an inhomogeneous
semiconductor heterostructure grown by liquid-phase epi-
taxy. An Al0.5Ga0.5As0.05Sb0.95layer with a thickness of
roughly 3.3 ?m was grown on a cleaned surface of a ?001?
GaSb substrate at 600°C. Another layer with the composi-
tion In0.22Ga0.78As0.19Sb0.81, whose thickness was tentatively
of the order of 1?m according to the epitaxial technology,
was created above that layer.
The measurements of the angular distribution of the dif-
fracted radiation were performed on a Topo high-resolution
double-crystal diffractometer from the Japanese manufac-
turer Rigaku. The double-crystal camera was combined with
an RU-200 x-ray tube. The primary beam was collimated
and monochromatized using the ?440? reflection from a per-
fect germanium?001? crystal in Cu K?1radiation. The asym-
metry factor had the value b?0.095, which ensured an an-
300Tech. Phys. 44 (3), March 1999S. G. Podorov and V. I. Punegov
gular divergence of the primary beam smaller than 1?. The
x-ray diffraction photograph of the heteroepitaxial structure
was taken at the symmetric ?006? reflection with a Bragg
angle equal to 49.46°.
The diffraction reflection curve has a form which is typi-
cal of inhomogeneous epitaxial structures with a positive lat-
tice strain gradient ?Fig. 1?. The behavior of the oscillations
on the experimental diffraction reflection curve points to the
presence of a constant or nearly constant strain gradient in
the Al0.5Ga0.5As0.05Sb0.95layer. Linear distribution of the
components of the solid solution across the thickness of this
layer apparently took place during epitaxial growth. The
mean interplanar distance of the reflecting atomic planes in
the upper In0.22Ga0.78As0.19Sb0.81layer is smaller than the
corresponding value for the GaSb substrate. Therefore, the
diffraction peak from the In0.22Ga0.78As0.19Sb0.81layer is in
the large-angle region at a distance of 200? from the sub-
The initial computed diagnostics approximation was ob-
tained using a procedure for a model of a crystal with linear
variation of the interplanar distance across its thickness.12
The static Debye–Waller factor was set equal to unity.
The results of the calculations performed by the method
described above are shown in Figs. 1–3. In Fig. 1 the theo-
retical diffraction reflection curve is represented by the
dashed line. As expected, the Al0.5Ga0.5As0.05Sb0.95epitaxial
layer has an essentially constant crystal-lattice strain gradient
?Fig. 2?. The upper In0.22Ga0.78As0.19Sb0.81layer is also inho-
mogeneous, its thickness being less than 1 ?m. It should also
Al0.5Ga0.5As0.05Sb0.95layer are separated by a transition re-
gion with a thickness of the order of 0.3 ?m, which is
formed either during the growth process or as a result of
Al0.5Ga0.5As0.05Sb0.95layer and the substrate is fairly sharp.
The results obtained during the calculations for the static
Debye–Waller factor indicate that the regions near the het-
erointerfaces and the surface of the sample have the highest
defect density ?Fig. 3?. These structural features were also
observed for other multilayer systems analyzed by various
methods.8–10,13The mean value of the static Debye–Waller
factor of the Al0.5Ga0.5As0.05Sb0.95layer is roughly equal to
0.8. Essentially the same crystal perfection was exhibited by
a previously investigated graded AlGaAs layer grown by
metalorganic vapor-phase epitaxy.9,10According to the re-
sults obtained, the upper In0.22Ga0.78As0.19Sb0.81layer has a
structure with a higher defect density. The strong jump in the
degree of amorphization on the boundary between this layer
and the Al0.5Ga0.5As0.05Sb0.95layer is due to the large mis-
match between the lattice parameters of these two com-
pounds. As a result, the strong lattice-parameter mismatch
leads to relaxation processes, which partially or completely
eliminate the tetragonal strain in the multilayer structure.
This, in turn, is accompanied by additional defect formation.
Thus, one of the simplest methods for numerical solution
of the inverse problem of diffraction in a distorted crystal
FIG. 1. Calculated ?1, dashed line? and experimental ?2, solid line? diffrac-
Al0.5Ga0.5As0.05Sb0.95/?001? GaSb heterostructure.
Al0.5Ga0.5As0.05Sb0.95/?001? GaSb heterostructure.
FIG. 3. Variation of the static Debye–Waller factor across the thickness of
an In0.22Ga0.78As0.19Sb0.81/Al0.5Ga0.5As0.05Sb0.95/?001? GaSb.
301Tech. Phys. 44 (3), March 1999 S. G. Podorov and V. I. Punegov
structure has been developed. The method permits obtaining
information on the distribution of crystal-lattice strains and
the degree of amorphization within a fairly short time. Since
these structural characteristics are related to the composition
of the semiconductor structure investigated, the method
opens up additional possibilities for studying relaxation and
defect-formation processes in epitaxial multilayer structures
as a function of the growth technology.
Because of the features of the method developed, all the
solutions were obtained in the form of continuous functions.
The method for solving the problem is recurrent; therefore, a
good choice for the initial approximation permits significant
shortening of the time for the calculation procedure. In addi-
tion, there is the problem of nonuniqueness of the solution of
the inverse problem.7For this reason, careful selection of the
starting approximation and utilization of a priori information
regarding the technology used to fabricate the samples are
important steps in structural diagnostics. The algorithm for
the method developed calls for calculating Fourier integrals,
and large layer thicknesses require an increase in the angular
interval during the calculations.
Finally, we note the following fact: the algorithm devel-
oped is relatively simple and does not require lengthy com-
plicated calculations. The numerical solution of the inverse
diffraction problem on a personal computer with a
Pentium-90 processor for 10 iterations takes approximately
10 min when the file for different depths contains 1200 val-
ues and the file for different angles contains 400 values.
We thank V. A Kusikov for providing the experimental
1J. Burget and D. Taupin, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr.,
Theor. Gen. Crystallogr. 24, 99 ?1968?.
2J. Burget and R. Collela, J. Appl. Phys. 40, 3505 ?1969?.
3A. Fukuhara and Y. Takano, Acta Crystallogr., Sect. A: Cryst. Phys.,
Diffr., Theor. Gen. Crystallogr. 33, 137 ?1977?.
4A. M. Afanasev, M. V. Kovalchuk, E. K. Kovev, and V. G. Kohn, Phys.
Status Solidi A 42, 415 ?1977?.
5R. N. Kyutt, P. V. Petrashen’, and L. M. Sorokin, Phys. Status Solidi A
60, 381 ?1980?.
6B. C. Larson and J. F. Barhorst, J. Appl. Phys. 51, 3181 ?1980?.
7A. V. Goncharski?, A. V. Kolpakov, and A. A. Stepanov, Inverse Prob-
lems of X-Ray Diffractometry ?in Russian?, Latvian University, Riga
?1992?, 181 pp.
8S. G. Podorov, V. I. Punegov, and V. A. Kusikov, Fiz. Tverd. Tela ?St.
Petersburg? 36, 827 ?1994? ?Phys. Solid State 36, 454 ?1994??.
9V. I. Punegov and N. N. Faleev, Fiz. Tverd. Tela ?St. Petersburg? 38, 255
?1996? ?Phys. Solid State 38, 143 ?1996??.
10V. I. Punegov, K. M. Pavlov, S. G. Podorov, and N. N. Faleev, Fiz. Tverd.
Tela ?St. Petersburg? 38, 264 ?1996? ?Phys. Solid State 38, 148 ?1996??.
11D. Taupin, Bull. Soc. Fr. Mineral. Cristallogr. 87, 469 ?1964?.
12A. V. Kolpakov and V. I. Punegov, Poverkhnost’ No. 3, 82 ?1988?.
13K. M. Pavlov, V. I. Punegov, and N. N. Faleev, Zh. E´ksp. Teor. Fiz. 107,
1967 ?1995? ?JETP 80, 1090 ?1995??.
Translated by P. Shelnitz
302 Tech. Phys. 44 (3), March 1999S. G. Podorov and V. I. Punegov