# Iterative solution of the inverse problem of dynamic diffraction in inhomogeneous crystals

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- Max Planck Institute of Quantum Optics, 06/2002, Degree: Dr. rer. nat., Supervisor: Prof. Dr. Eckhart Foerster

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Iterative solution of the inverse problem of dynamic diffraction in inhomogeneous

crystals

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?

1. INTRODUCTION

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-

state physics.

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

Takagi–Taupin equations

d

dzE0?z??i?0E0?z??i??gU*?z?Eg?z?,

?

d

dzEg?z??i????0?Eg?z??i?gU?z?E0?z?,

?1?

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

?0???0/???0?;

?g,?g???g,?gC/????g,0??;

??2???0?sin?2?0????/???0?.

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)

U?z??B?z?exp??iF?z??,

F?z??g•u?z???

0

z

f?x? dx,

f?z???2??d?z?/d2.

?2?

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

B?z??e?W?z?.

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

d

dzRg?z,????i?gE?z??i???f?Rg?z,??

?i??gE?z?Rg

2?z,??

?3?

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

PROBLEM

Letusconstructthefunction

Rk?U?(?)

?i?g?0

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

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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

?Rg?U?????Rg?U????*?Ig???,

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:

?4?

A?U?V??Rg?U??*?aRk?V??Rg?U??aRk?U??,

where a is a parameter.

Equation ?4? can then be written in the following form:

?5?

A?U?U?Ig???,

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

equation ?6?:

?6?

A?U?n?1??U?n??Ig???.

Hence we have

U?n?1??z??A?1?U?n??Ig???

or

U?n?1??z??U?n??z??

1

2?ai?g

??

??

Ig?Ig?U?n??

Ig?U?n??

Rg?U?n??e?i?zd?.

?7?

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

B?n?1??z???

?aIg?U?n????Rk?U?n???e?i?zd??.

1

2??g?

????Ig?Ig?U?n???Rg?U?n??/

?8?

The lattice strain profile is found from the iteration for-

mula

f?n?1??z??

d

dzF?n?1??z?

?Re?i?

dU?n?1??z?

dz ?U?n?1???,

?9?

where the derivative has the form

dU?n?1??z?

dz

?

?1

2??g?

????Rk?U?n??

??Ig?Ig?U?n???Rg?U?n??/

?aIg?U?n????e?i?zd?.

?10?

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

function

?B?z????

?f?z????

?xexp???z?x?2

?xexp???z?x?2

??B?x? dx,

??f?x? dx.

?11?

?12?

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

taken.

6. DETERMINATION OF THE STRUCTURAL

CHARACTERISTICS OF AN INHOMOGENEOUS

In0.22Ga0.78As0.19Sb0.81/Al0.5Ga0.5As0.05Sb0.95/„001… GaSb

HETEROEPITAXIAL SYSTEM

The computational iterative procedure developed is ap-

plicable to the structural diagnostics of an inhomogeneous

In0.22Ga0.78As0.19Sb0.81/Al0.5Ga0.5As0.05Sb0.95/?001?

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-

GaSb

300Tech. Phys. 44 (3), March 1999S. G. Podorov and V. I. Punegov

Page 3

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-

strate peak.

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

benotedthat theupper

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

layerandthe graded

self-diffusion.

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.

Theboundary betweenthe

7. CONCLUSION

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-

tionreflectioncurves of

Al0.5Ga0.5As0.05Sb0.95/?001? GaSb heterostructure.

anIn0.22Ga0.78As0.19Sb0.81/

FIG.

Al0.5Ga0.5As0.05Sb0.95/?001? GaSb heterostructure.

2. Strainprofile

??d(z)/d?

of anIn0.22Ga0.78As0.19Sb0.81/

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

Page 4

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

results.

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Translated by P. Shelnitz

302 Tech. Phys. 44 (3), March 1999S. G. Podorov and V. I. Punegov