X-Ray Diffraction Analysis of III-V Superlattices:Characterization, Simulation and Fitting

Enlong Liu Xiangyu Wu ∗

Abstract

Three samples of III-V semiconductor superlattice (SL) are investigated by X-ray diffraction(XRD). The XRD curves exhibit a number of features indicative of SL properties, includingzero-order diffraction peak of both SL and the layer between SL and substrate, and a series ofsatellite peaks. Detailed models for each sample are established to simulate XRD results and byfitting simulation curves with experimental ones, the composition and period of each layer inSL is derived, which is nearly in consistent with the results given by theoretical calculation andthe measurement by Transmission Electron Microscopy (TEM). XRD turns out to be a simple,reliable and fast characterization method for SL researches.

1 Introduction

III-V semiconductor superlattice (SL) has numerous applications in electronic [1–3] and op-toelectronic [4,5] devices after its electronic structure was fully understood [6] and the depositionmethods become mature. Its heterostructure is also a model to understanding the growthmechanism and interfaces of semiconductor alloys in different ways, such as Chemical VapourDeposition (CVD) and Molecular Beam Epitaxy (MBE). However, a little fluctuation in growthenvironment like temperature and the different properties at interfaces may cause a huge devi-ation in composition and period of SL structure and hence the device behaviours. For example,the composition of In0.53Ga0.47As grown by MBE turns out to be In0.6Ga0.4As in real sample.As the request of real-time monitoring of the growth progress, a fast and reliable way to char-acterize the deposited SL is needed. Transmission Electron Microscopy (TEM) is widely usedin measuring the period of SL. But the composition of SL can not be read from TEM figuresand preparing samples for TEM measurement is destructive to origin samples and the mosttime-consuming part in a series of experiments.

X-Ray Diffraction (XRD) has been used to investigate crystal properties in powders for along time and to analyse thin film [7,8], III-nitrides [9] and SL in recent years by applying the con-cept of X-Ray Reflectivity (XRR) [7,10], since the structure of SL is like the amplification of crystalplanes, leading to periodicity in rocking curves. These X-Ray rocking curves provide informa-tion on strain, layer thickness, compositions and diffusion profiles in the structure assessmentof SL [11]. Unlike that of TEM, the sample of XRD is easy to prepare. The measurements can becompleted in situ during SL deposition onto wafer and take less time than TEM. Because of itsnon-destructive and highly efficient microstructural characterization, XRD has already been

∗These two authors contributed equally to this report.

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used to study several kinds of SL structures, such as InGaAs/GaAs [12] and InGaAs/InP [13].However, these works assumed that the SL layers in sample are in the definite composition andthickness under ideal processing conditions, which is obviously not the case. And their worksmostly concentrated on the strain in SL layer and the growth mechanism. In our project, weused XRD to study SLs with more complex structures and analysed the rocking curves basedon theoretical calculations and models with parameters optimized by simulations and fittings.This report will first introduce the relative theory of XRD and its application in SL, then thecharacterization, simulation and fitting of three samples are summarized. At last, we will com-pare the results from XRD to those from TEM to verify the reliability of XRD measurement inSL studies.

2 XRD

2.1 Principle

Scattering theory describes the interaction between light and matter. Particles can absorblight and be excited into an higher energy level, and then relax to the ground state by re-emission of light, this phenomenon is called scattering. The scattering process can changethe direction and frequency of light. Different scattering mechanisms are shown in Fig.1. InRayleigh scattering, the frequency of light is not changed, while in Raman scattering, frequencyis increased (anti-Stokes scattering) or decreased (stokes scattering). Raman scattering is veryuseful in analysing molecular structure. But in XRD crystal analysis, Rayleigh scattering is de-tected, Raman scattering only decreases the intensity of XRD signal. The most important the-

Figure 1: Band diagrams for Raman and Rayleigh scattering

ory for XRD is the Bragg‘s Law. William Lawrence Bragg and William Henry Bragg observedthat when X-ray strikes crystals, the reflected X-ray with produce special patterns. At certainwavelength and incident angel, a strong reflection peak was observed. This phenomenon wasdescribed by Bragg‘s law. As shown in Fig.2, two beams with identical wavelength and phaseapproach a crystalline solid and are scattered off by two different atoms within it. The lowerbeam traverses an extra length of 2d sin Θ. Constructive interference occurs when this lengthis equal to an integer multiple of the wavelength of the radiation.

In the famous Bragg‘s equation,nλ = 2d sin Θ (1)

λ is the wavelength of incident beam, Θ is the incident angle, n is an integer, d is the distancebetween the two atom plane. When this equation is satisfied, constructive interference will

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occur, leading to Bragg peaks. Thicker material with more atom planes will produce narrowerand stronger peaks. The basic scattering unit of a crystal is its unit cell.

Figure 2: Schematic drawing of Bragg‘s Law

As shown in Fig.3(a), XRD consists of a X-ray tube which produces X-ray source, a planefor specimen and a detector which collects the scattered X-ray signal. In Fig.3(b), ω is theangle between the emitter and substrate. 2Θ tunes the angle between emitter and detector.Ψ represents the vertical rotation of substrate plane, while Φ is the horizontal rotation of thesubstrate plane. At last, the x− y− z coordinates control the position of the specimen.

(a) (b)

Figure 3: Structure of XRD system: (a) Schematic drawing of the XRD machine; (b)Explanation for different parameters in XRD measurement

Two commonly used scan methods are ω-scan and ω− 2θ coupled scan, shown in Fig.4(a)and (c). In a ω scan, 2θ is fixed. In a coupled scan, ω and θ move simultaneously with therelationship ω = 2θ + offset. The triple axis diffractometer observes all peaks within a certainrange like Fig.4(d), while ω scan only observe one special diffraction peak as in Fig.4(c). Withω scan, we can measure tilt independent of strain (dilation) and get defect information for eachindividual layer, but tilt and strain could not be independently resolved using a single double-axis rocking curve. A single coupled scan can resolve differences in the d-spacing values ofthe crystallographic planes (d-spacing corresponds to mismatch, composition, relaxation). Acommon strategy is to collect an ω − 2θ scan, identify peak positions, then collect the rockingcurve for each diffraction peak.

2.2 Superlattice Diffraction

The greatest use of XRD in industry is the characterization of epitaxial structures on com-pound semiconductors. The composition of ternaries, mismatch of quaternaries, mis-orientation,layer thickness, tilt, relaxation, indications of strain, curvature and stress, and area homogene-ity have important influence on the performance of III-V and II-VI semiconductors. With XRD,

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(a) (b) (c) (d)

Figure 4: (a) Schematic drawing for ω scan; (b) Curve of Si by XRD ω scan ; (c) Schematicdrawing for ω− 2θ scan ; (d) Curve of SiGe double layer by XRD ω− 2θ coupled scan

we can analyse all these parameters in a cheap, fast, non-destructive way. Of all these param-eters, mismatch is the most important one, because it gives information not only about thestrain, but also about the lattice constant and composition, which is further linked with theenergy bandgap. For the same reason, layer thickness is also very important, due to quantumsize effect. So in Rocking curve, we can always visualize two peaks or more, which is causedby tilt or mismatch between the substrate and superlattice epitaxial layers. Diffraction willalso occur between different interfaces, so many small peaks will also be observed which givesinformation about the different layer thickness.

As shown in Fig.5, according to Bragg‘s law, constructive diffraction will happen when θ

and d satisfies certain condition,λ = 2dhkl sin θ (2)

where d is the inter-plane distance, θ is the Bragg angle. In crystal, the basic diffraction unit isunit cell. For III-V semiconductor crystal with blende cubic structure (cubic crystal), we alsohave

1d2

hkl=

h2 + k2 + l2

a2 (3)

For (004) plane used in the test, we have

d004 =a4

(4)

, in which a is the lattice constant. Combine these equations, we can get the relation betweenwavelength of X-ray and lattice constant as follow:

λ =a2

sin θ (5)

According to this equation, lattice constant a can be calculated, which gives information aboutmismatch. Moreover, Vegard‘s Law [14] gives out

aInxGa1−xAs = xaInAs + (1− x)aGaAs (6)

, with which the composition of the SL can be calculated. To extract the composition informa-tion from XRD curves is one of our main targets.

The second target is to extract the thickness of each layer in the SL structure and verifyresults with TEM image. According to Eq.(1), we get the expressions for two neighbouringBragg peaks,

Nλ = 2Λ sin θn (7)

(N − 1)λ = 2Λ sin θn−1 (8)

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Figure 5: Drawing for the mechanism of extracting mismatch and composition from XRD

. Combine these two equations, the equation for thickness calculation can be derived as below,

λ = 2Λ(sin θn − sin θn−1) (9)

2 sin θn − 2 sin θn−1

λ=

1Λ

(10)

. To eliminate errors, the final and most important equation is derived by averaging over thepositions of satellite peaks of order n:

2 sin θn − 2 sin θSL

λ= ± n

Λ(11)

, where Λ is the thickness of SL period, λCuKα = 0.15405nm is the wavelength of incident X-ray,θn is the n th-order peak in rocking curve, θSL is the zero-order peak.

(a) (b)

Figure 6: Schemes indicating the calculation of layer thickness: (a) Example of XRD rockingcurves [13]; (b) X-ray diffraction in SL structure

3 Results and Discussions

3.1 Sample Structure

All three samples are deposited by MBE in different conditions, whose structures are shownin Fig.7. Every sample has a InP(001) substrate, which is 600¯m thick, and a InAlAs buffer layerdeposited on top of a Si wafer sequentially. The number of repeat unit in their superlattice(SL) is 5. The difference of the structure lies in SL materials. For sample 1 (S1) and sample2 (S2), their SL layers are made of InP and InGaAs alternatively, and the thickness of eachlayer of SL in S2 is twice of that in S1, which is determined by the time of MBE depositionprocess. While the SL layers of sample 3 (S3) is made of InGaAs and InAlAs respectively,and the thickness of InGaAs is the same as that in S2, but for InAlAs all information is totally

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unknown. Furthermore, the composition of both InAlAs buffer layer and repeat unit in SL arealso needed to be determined. These samples can be used in XRD experiment directly insteadof preparation like TEM samples.

Figure 7: Schematic drawings for three samples: (a) S1; (b) S2; (c) S3

3.2 XRD Experiments Results

XRD measurements are conducted on XPert Pro MRD machine made by PANalytical. Weused double-axis ω − 2θ mode in XRD measurement to characterize all three samples. TheX-ray incident plane on the samples has been optimized by justifying Φ and Ψ offset to getmaximum signal intensity. The incident angle range for measurement ω ' 4◦, and step size is0.00004◦. The result curves for three samples are shown in Fig.8, with modification in ampli-tude and substrate peak offset for clear comparison.

Figure 8: XRD curves for S1. S2 and S3

At the first glance of Fig.8, every curve has two big peaks and many satellite peaks, andthe distance between satellite peaks in S2 is smaller than that in S1. These peaks contain all theinformation about the composition and thickness in buffer layer and SL layers in each sample.What we will do is to derive all the parameters of these samples from these XRD curves afteridentifying the peak belongings, calculating theoretically and simulating. At this stage, only

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the greatest peak in each sample can be determined as the InP substrate peak, the others willbe studied in the following section.

3.3 Curves Analysis and Simulation

3.3.1 Peaks Belonging

XRD curves are quite complicated with many peaks and fringes as above. It can only besure that the highest peak is the substrate peak, because substrate has the largest thickness andhence the highest diffraction intensity. XRD analysis is based on the understanding of XRDcurves, without which it‘s impossible to do calculation or simulation. So in this section wefirst made a simulation of only the substrate and buffer layer to understand the buffer layer‘scontribution on total curve. Then we also made a simulation with only substrate and SL inorder to exclude the contribution from buffer layer and analyse how SL influences the totalcurve. The simulation results are shown below in Fig.9.

(a)

(b)

Figure 9: Separate simulation results: (a) InAlAs buffer layer simulation; (b) InGaAs/InPsuperlattice simulation

The highest peak labelled by purple arrow appears in both simulations, meaning that thispeak is produced by the substrate diffraction. This Bragg peak is decided by λ = aInP

2 sin(θ −

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tilt). In Fig.9(a), the second highest peak labelled by green arrow only appears in the bufferlayer simulation, meaning that this peak has nothing to do with SL and substrate, so it shouldbe the InAlAs buffer layer peak, because the buffer layer has the second highest thickness, sothe peak intensity is only next to the substrate peak. Many fringes are observed around bufferlayer peak in Fig.8(a), which are caused by the constructive diffraction of reflection betweenthe top surface and bottom surface of the buffer layer, the distance between these fringes isdecided by the thickness of the buffer layer. These fringes are also observed in the circled areain the experimental curve, which have good consistence with the simulation.

Fig.9(b) comes to the contribution from SL to total curve in . The most important peakis labelled by the red arrow. This peak is the 0-order peak of SL layer. This peak is formedby mismatch, reflecting the difference in the lattice constant of SL layer, this leads to differ-ence in Bragg angles. If this peak appears at the left side of the substrate peak, it means thataInGaA > (aInP) and SL layer is under compressive strain and vice versa. Misorientation canalso influence this peak. The smaller peaks labelled by black arrows are satellite peaks formedby the constructive interracial diffraction of SL, which give information about SL layer thick-ness. If the uniformity of different SL layers are bad, these peaks will be broadened as aninstruction of interlayer uniformity. Unfortunately, no numerical results of thickness variationcan be extracted at present. Moreover, there are always three smaller peaks between every twoneighbouring satellite peaks in the simulation curve. Its number N gives information aboutthe number of SL period by N + 2. So three small peaks means the SL period is repeated byfive times, which is coincided with our sample structures.

3.3.2 Theoretical Calculation

S1 After analysing the source of all the peaks in XRD curve, the theoretical calculation ofall the parameters can be carried out. For the sake of redundancy, we will only calculate S1parameters in detail.

The peaks labelled by the red arrows in Fig.10 are the fringes of the InAlAs buffer layer. Ac-cording to Eq.(10), we get the buffer layer thickness of 117nm. SL period thickness is calculatedwith the angles labelled by blue arrows by using

2 sin θ4 − 2 sin θSL

λ= ± 4

Λ(12)

, where θ4 = 32.29795, θSL = 31.67345, λ = 0.15405nm. And we get Λ ≈ 34.2nm. Also wenotice that the even order peak is much weaker than the odd order peak, indicating that thethickness of the two layers in SL are quite similar. So we assume that the thickness of eachlayer is 17.1nm.

From the deviation between the substrate peak and 0-order peak, the mismatch and com-position of SL can be derived. Take θs = 31.69545 and θSL = 31.67345 into

offset = θs − θInP (13)

andλ =

a2

sin(θSL − offset) (14)

where θInP = 31.668 is already known, this leads to aInGaAs = 5.874. According to Eq.(6)and lattice constant as aInAs = 6.058 and aGaAs = 5.623, the composition of InGaAs in SL isx = 0.577, i.e. its formula In0.577Ga0.423As/InP.

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However, this calculation is based on a very simplified model, where the influence of re-fractive index isn‘t taken into consideration. And in the experiments, the contribution of mis-orientaion on the separation between 0-order peak and substrate peak fails to be eliminated,which might be important. Kinetic model should be used if the precise value from calculationis desired, though the value above is good enough to be used as initial for further simulationand fitting.

Figure 10: Peak analysis for S1: Red arrows indicate the peaks originated from buffer layer;blue arrows show the different orders of peak originated from SL, the 2nd order peak dis-appears because of destructive interference

S2 With the in-detail analysis for peak belongings in S1, it is natural to use the same method-ology to study peaks in S2 curve. By splitting the whole sample structure into two part, sub-strate with buffer layer only and substrate with SL only, it is easy to find out which peak comesfrom where in S2 curve. The results are shown in Fig.11. The initial values for simulation of S2come from two parts. The first one is from known conditions listed in Section 3.1, saying thatthe thickness of SL in S2 is twice as that in S1. Besides, as we assume the two layers in SL pe-riod are in same thickness, we can use 34, 2nm as the initial thickness for InGaAs layer, whichis given out from 17.1nm multiplied directly by 2 in Table 1. On the other hand, the period ofSL in S2 can be calculated using Equation after identifying the peaks in Fig. To minimize theerror, the angle value of 1st order and 7th order of peak are used, leading to Λ = 70nm. Thus,the thickness of InP is 70− 34.2 = 35.8nm. The other calculated results are listed in Table 1.

S3 The curve of S3 is more complicated, but peak identification and calculation is nearly thesame as that in S2. Based on what we knew for this sample, i.e. the thickness of InGaAs isthe same as S2, it is easy to know its initial value as 34.2nm. According to Fig.12, the intensityof the even order of satellite peaks are still very high, indicating the different thickness of thetwo layers in one SL period. Furthermore, this SL period can also be calculated with Equation,which is 53.6nm. So the thickness of InAlAs is 53.6 − 34.2 = 19.4nm. The composition ofInGaAs is given by using the angle value of 0 order peak and Equation. As for the compositionof InAlAs in SL, there is no 0 order peak for this layer, meaning that it has the same latticeconstant with the substrate InP. With Equation, the composition of InAlAs can be calculatedas well, which is listed in Table 1.

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Figure 11: Peak analysis for S2: Red arrows indicate the peaks originated from buffer layer;blue arrows show the different orders of peak originated from SL, the 2nd order peak dis-appears because of destructive interference

Figure 12: Peak analysis for S3: Red arrows indicate the peaks originated from buffer layer;blue arrows show the different orders of peak originated from SL, the even order peaksappear at this time

Table 1: Thickness and Composition of three samples given by theoretical calculation

Sample

S1 S2 S3

Substrate-InP 600µm

Layer-InAlAs 117nm 126nm 135nm

Substrate

Layer 1 InP 17.1nm InP 35.8nm InGaAs34.2nm

In-0.556

Layer 2 InGaAs17.1nm

InGaAs34.2nm

InAlAs19.4nm

In-0.577 In-0.572 In-0.52

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Table 2: Thickness and Composition of three samples given by simulation and fitting

Sample

S1 S2 S3

Substrate-InP 600µm

Layer-InAlAs 135nm, In-0.59 132nm, In-0.52 135nm, In-0.52

Substrate

Layer 1 InP 17.9nm InP 36.0nm InGaAs29.7nm

In-0.60

Layer 2 InGaAs17.0nm

InGaAs34.5nm

InAlAs21.7nm

In-0.60 In-0.60 In-0.52

3.3.3 Simulation results and discussions

With all the initial values in thickness and composition for all three sample, we use sim-ulation and fitting to finalize the accurate value of these parameters. The software packagesnamed XPert Epitaxy and XPert Smoothfit provided by PANalytical are used for simulationand fitting respectively. The simulation curves and fitting results are discussed below.

In Table 2 are listed all data derived from XRD curves under the analysis flow from theoret-ical calculation to simulation and fitting. It is clear that the thickness in each sample satisfiesthe relations we have already talked about in Section 3.1 and the composition for InGaAs inSL keeps constant for three samples, meaning that the deposition is stable for different sampleprocesses. As mentioned above, the simulation curve for S3 is not quite identical with the ex-periment one, it needs further confirmation and modification after its comparison with TEMresults, as well as the other two samples.

Fig.13(a)-(c) show simulation curves compared with experimental curves of three samplesafter horizontal translation to the curves for better correspondence. As it is shown, all simula-tion curves are in consistent with experimental ones very well, indicating the great similarityof our established fitting models with the real samples.

4 Comparison with TEM Results

Even though the composition and thickness of the samples are already got from XRDcurves, it is necessary to verify them. To do so, TEM pictures are used, for the thickness ofdifferent layers can be read out directly. Fig. shows a series of TEM pictures for three samples.Fig.14(a)-(c) give out the thickness of InAlAs buffer layer, which is 137nm, 140nm and 136nmfor S1, S2 and S3 respectively. Fig.14(d)-(f) show the layer thickness in SL part of all samples.For S1, its InP layer is 18nm thick and InGaAs is 16nm. In S2 the InP layer is 38nm and InGaAslayer is 33nm, both of which are nearly two times of those in S1. These data indicate the ac-curacy of MBE deposition process, so does the 32nm-InGaAs layer in S3, which is nearly thesame as 33nm in S2. The totally unknown InAlAs layer in S3 is shown to be 20nm in thickness.The limitation of TEM is in its inability to get composition of different layers in each sample,which is the superiority of XRD.

Listed in Table 3 are the data from TEM and XRD. The composition information can notbe read from TEM figures, but is uniquely given by XRD curves. Though data of S1 and S2

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Figure 13: Comparison between simulation curves and experimental curves for (a) S1, (b) S2 and (c) S3

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Figure 14: TEM pictures with thickness labels for three samples: (a) S1 bufferlayer; (b) S2 buffer layer; (c) S3 buffer layer; (d) S1 SL layers; (e) S2 SL layers;(f) S3 SL layers.

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Table 3: Comparison between results given by XRD and TEM

Sample Part Material XRD TEM Error

S1

Layer InAlAs 135nm 137nm −2nm

SL

InP 17.9nm 18.0nm −0.1nm

InGaAs17.0nm 16.0nm +1.0nm

0.6 NA NA

S2

Layer InAlAs 132nm 140nm −8nm

SL

InP 36.0nm 38.0nm −2nm

InGaAs34.5nm 33.0nm +1.5nm

0.6 NA NA

S3

Layer InAlAs 135nm 136nm −1nm

SL

InGaAs29.7nm 32.0nm −2.3nm

0.6 NA NA

InAlAs21.7nm 20.0nm +1.7nm

0.52 NA NA

show a good consistency between XRD and TEM, data of S3 has relatively large error, whichcorresponds to the fact of deviation in simulation curve to experimental one in Fig.(c). It maycome from the more complex structure of S3 with too many parameters and the imperfectsimulation software algorithm. Besides, the interlayer between SL and buffer layer is not takeninto consideration for all three samples, which may also be the source of error. The influencefrom the interlayer is being researched in the following work.

All in all, it is obvious from the comparison that the errors are acceptable and within thepermissible range, indicating that XRD characterization, simulation and fitting has high relia-bility in deriving the thickness of different layers and provides extra informations which cannot be given by TEM, i.e. compositions. It makes XRD not only a complement for TEM, but aintact method for SL studies.

5 Conclusion

We have carried out XRD studies on superlattice samples with different compositions andperiods. Based on the information derived from XRD rocking curves, three models were estab-lished and simulated. The fitting results of all three models not only gave information whichTEM could not, but also corresponded well with data already given by TEM figures, indicat-ing the reliability and accuracy of XRD measurement in superlattice structures. With its non-destructive property and high efficiency in conducting experiments and results derivation, theapplications of XRD in superlattice fields can be foreseen.

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6 Acknowledgement

The authors would like to thank Dr. Clement Merckling for the teaching, instruction andorganization throughout the whole project, and Dr. Weiming Guo for his fruitful discussionsand practical assistance, and the group leader Prof.Dr. M. Caymax.

References

[1] J. Tsai, et al., Applied Physics Letters 96, 063505 (2010).

[2] J. Gu, et al., Electron Devices Meeting (IEDM), 2012 IEEE International (2012), pp. 23.7.1–23.7.4.

[3] M. Sugiyama, et al., Optoelectronic and Microelectronic Materials Devices (COMMAD), 2012Conference on (2012), pp. 125–126.

[4] G. Scamarcio, et al., Science 276, 773 (1997).

[5] M. Courel, J. Rimada, L. Hernandez, Progress in Photovoltaics: Research and Applications 21,276 (2013).

[6] D. Smith, C. Mailhiot, Rev. Mod. Phys. 62, 173 (1990).

[7] M. Birkholz, Thin film analysis by X-ray scattering (wiley-vch, 2006).

[8] P. Fewster, Semiconductor Science and Technology 8, 1915 (1993).

[9] M. Moram, M. Vickers, Reports on Progress in Physics 72, 036502 (2009).

[10] D. Bowen, B. Tanner, High Resolution X-Ray Diffractometry And Topography (Taylor & Fran-cis, 1998).

[11] V. Speriosu, T. Jr.Vreeland, Journal of Applied Physics 56, 1591 (1984).

[12] Z. Ming, et al., Applied Physics Letters 66, 165 (1995).

[13] D. Cornet, R. LaPierre, D. Comedi, Y. Pusep, Journal of Applied Physics 100, 043518 (2006).

[14] A. R. Denton, N. W. Ashcroft, Phys. Rev. A 43, 3161 (1991).

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