study of threading dislocation density reduction in algan ...2of8s. lazarev et al. dislocation...

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J. Appl. Cryst. (2013). 46 doi:10.1107/S0021889812043051 1 of 8

Journal of

AppliedCrystallography

ISSN 0021-8898

Received 26 July 2012

Accepted 15 October 2012

# 2013 International Union of Crystallography

Printed in Singapore – all rights reserved

Study of threading dislocation density reduction inAlGaN epilayers by Monte Carlo simulation of high-resolution reciprocal-space maps of a two-layersystem

S. Lazarev,a M. Barchuk,b,a S. Bauer,a* K. Forghani,c V. Holý,b F. Scholzc and T.

Baumbacha

aKarlsruhe Institute of Technology (KIT)/Synchrotron Facility ANKA, 76344 Eggenstein-Leopold-

shafen, Germany, bCharles University in Prague, Faculty of Mathematics and Physics, Ke Karlovu 5,

12116 Prague 2, Czech Republic, and cInstitute of Optoelectronics, University of Ulm, Albert-

Einstein-Allee 45, 89081 Ulm, Germany. Correspondence e-mail: [email protected]

High-resolution X-ray diffraction in coplanar and noncoplanar geometries has

been used to investigate the influence of an SiNx nano-mask in the reduction of

the threading dislocation (TD) density of high-quality AlGaN epitaxial layers

grown on sapphire substrates. Our developed model, based on a Monte Carlo

method, was applied to the simulation of the reciprocal-space maps of a two-

layer system. Good agreement was found between the simulation and the

experimental data, leading to an accurate determination of the dislocation

densities as a function of the overgrowth layer thickness. The efficiency of the

SiNx nano-mask was defined as the ratio of the TD densities in the AlGaN layers

below and above the mask. A significant improvement in the AlGaN layer

quality was achieved by increasing the overgrowth layer thickness, and a TD

density reduction scaling law was established.

1. IntroductionNitride-based ultraviolet light-emitting diodes have been

investigated for their applications in high-density optical

storage media, biotechnology, air/water purification and

curing resins. By changing the Al composition, the cutoff

wavelength of AlGaN photoactive devices can cover the

spectrum from 200 to 360 nm at room temperature. However,

in comparison with GaN, AlGaN layers grown directly on

sapphire typically exhibit a large number of threading dislo-

cations (TDs), which mainly occur in the form of edge-type

TDs (Walker et al., 1996). The reduction of the number of

threading dislocations, which act as nonradiative recombina-

tion centres, is essential (Sugahara, Hao et al., 1998; Sugahara,

Sato et al., 1998) for the improvement of the performance of

ultraviolet light-emitting diodes.

Recently the optimization of Al0.2Ga0.8N layers directly

grown on sapphire by metal organic vapour phase epitaxy

(MOVPE) has been investigated (Forghani et al., 2011). The

measurement of the crystal quality of the AlGaN epilayers

was determined qualitatively from transmission electron

microscopy (TEM) micrographs. An improvement was found

by applying an in situ nano-masking technology with ultra-thin

SiNx interlayers. The edge-type dislocation density could be

significantly reduced by optimizing the growth parameters and

depositing the SiNx interlayer on a 150 nm Al0.2Ga0.8N inter-

layer grown on an AlN nucleation layer. The width of the

rocking curve of the asymmetric 10.2 reflection was used as a

good indicator for the reduced defect density in the

Al0.2Ga0.8N layer grown on top of the SiNx nano-mask

(Forghani et al., 2011).

A deeper insight into the growth mechanism and the defect-

reduction process has been obtained by exploiting weak-beam

dark-field micrographs in high-resolution TEM investigations

(Klein et al., 2011). These micrographs show a significant

difference of the dislocation densities below and above the

SiNx mask. However, this method does not enable the deter-

mination of the TD densities. To overcome the limitations of

the TEM investigation we have applied several approaches to

determine the dislocation densities from X-ray diffraction

data. In our previous work, the TD densities were determined

from simple equations based on the full width at half-

maximum of the rocking curve (Gay et al., 1953; Lazarev et al.,

2012).

Recently, Monte Carlo simulation has been used for the

modelling of reciprocal-space maps (RSMs) in coplanar and

grazing-incidence diffraction (GID) of GaN layers grown on

sapphire. By comparing the simulated and measured maps in

the diffuse part generated by defects, the TD densities have

been accurately determined (Barchuk et al., 2010, 2011).

In this paper, we demonstrate the possibility of using a

Monte Carlo method to compute more sophisticated

Al0.2Ga0.8N epilayer structures and to establish the scaling law

of the TD density reduction with overgrowth thickness. The

TD densities below and above the SiNx nano-mask have been

http://scripts.iucr.org/cgi-bin/cr.cgi?rm=pdfbb&cnor=he5567&bbid=BB16

derived from the Monte Carlo simulation with the goal of

estimating the SiNx nano-mask efficiency in defect reduction

in AlGaN epilayers.

2. Experiment

2.1. Sample description

In the present work we investigated seven samples of

c-plane epitaxial AlGaN structures grown on sapphire (00.1)

substrates with a miscut of 0.3� towards the a plane. Growth

was carried out in a low-pressure horizontal MOVPE reactor

(Aixtron AIX-200/4 RF-S) using the standard precursors

trimethylgallium (C3H9Ga), trimethylaluminium (C3H9Al)

and ammonia (NH3); the growth temperature was set to

1393 K. An Al content of 20% for the AlGaN samples was

derived from photoluminescence measurements (Neuschl et

al., 2010). The growth was initiated by an oxygen-doped AlN

nucleation layer (about 25 nm). After about 150 nm of AlGaN

growth, an SiNx nano-mask layer was deposited in situ by

flowing SiH4 and ammonia into the reactor for approximately

5 min. On the top of this layer, a

second AlGaN layer was deposited,

with thicknesses of 90, 290, 500, 1000,

1850, 2500 and 3500 nm corresponding

to samples A1–A7, respectively. The

layout of the AlGaN samples is shown

in the inset of Fig. 1(a) (left), where the

layer above the SiNx mask is termed

the overgrowth layer while the layer

below the mask is denoted the 150 nm

AlGaN interlayer.

2.2. Experimental setup and X-raymeasurements

High-resolution X-ray coplanar and

noncoplanar diffraction measurements

have been performed at the bending-

magnet single-crystal diffraction

(SCD) beamline at the synchrotron

facility ANKA in Karlsruhe in

Germany. Fig. 1(a) (left) shows the six-

circle diffractometer used to bring the

sample into the Bragg condition and to

measure symmetric and GID reflec-

tions. The diffractometer has four

degrees of freedom for the sample and

two degrees of freedom for the detec-

tor. All X-ray data were recorded with

a microstrip solid-state detector

(MYTHEN 1K, manufactured by

DECTRIS Ltd), having 1280 channels

with a channel size of 50 mm and apoint-spread function of one channel

(Bergamaschi et al., 2010).

Fig. 1(a) (middle) gives a schematic

representation of the coplanar diffrac-

tion geometry used to record RSMs of the symmetric 00.2,

00.4, 00.6 and 00.8 reflections, at an energy of 12 keV for all

samples. In this geometry the incident and the outgoing beams

lie in the plane perpendicular to the sample surface. In this

case the distance between the crystalline planes (hkl) parallel

to the surface, in this paper later referred to as ‘d spacing’, has

been determined.

As is illustrated in Fig. 1(a) (middle), �i and �f are definedas the angles between the incident and the outgoing beams

and the sample surface, respectively. The total angular range

of the scattering angle �f and the angular resolution are givenby the effective detector length (64 mm), the sample–detector

distance and the number of detector channels, as is shown in

Fig. 1(a) (middle).

The components of the scattering wavevector Q in the

radial and angular directions, qrad and qang, respectively, are

given by the following equations:

qrad ¼ ð2�=�Þ sin �f þ sin �ið Þ; ð1Þ

qang ¼ ð2�=�Þ cos�f � cos�ið Þ; ð2Þ

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2 of 8 S. Lazarev et al. � Dislocation density reduction in AlGaN epilayers J. Appl. Cryst. (2013). 46

Figure 1(a) The six-circle diffractometer at the SCD bending magnet beamline at ANKA (left), showing thelinear detector, the crystal analyser and a sample mounted on the Euler cradle stage. The inset showsthe layout of the Al0.2Ga0.8N samples. Schematic representations of the coplanar geometry used forsymmetric high-resolution X-ray diffraction (middle) and for the noncoplanar GID geometry (right).(b) Reciprocal-space map of the 00.2 reflection with two distinguishable peaks corresponding to AlNand AlGaN (left). The 00.8 reflection (middle) shows two AlGaN peaks, corresponding to theovergrowth layer and to the 150 nm AlGaN interlayer. The GID reflection 10.0 (right). All mapswere obtained using sample A4.

where � is the wavelength, qrad corresponds to the componentof Q along the surface normal and qang represents the in-plane

component of Q.

The variation of �i at the Bragg reflection was achieved bytilting the sample with respect to the incoming beam, which

leads to the acquisition of RSMs of the 00.2, 00.4, 00.6 and 00.8

symmetric reflections. In the RSM of the 00.2 reflection of

sample A4 (Fig. 1b, left), two main peaks are distinguishable:

an AlN peak coming from the nucleation layer and an AlGaN

peak, which shows diffuse scattering in the transverse direc-

tion (at qrad = 24.4 nm�1) due to the presence of dislocations.

In the RSM the intensity along the qrad direction at qang = 0

corresponds to the crystal truncation rod (CTR) where �i = �f.An additional streak (M) due to the crystal monochromator is

visible in the RSM of the 00.2 reflection.

Fig. 1(b) (middle) shows the highest measured Bragg

reflection, 00.8, of sample A4, where the two AlGaN peaks

designated as the AlGaN overgrowth layer and the 150 nm

AlGaN interlayer could be resolved. Such high-order reflec-

tions, which could readily be measured at the synchrotron

facility, are advantageous for resolving diffraction peaks

originating from only small differences in d spacing in the

growth direction. Besides the two AlGaN peaks, additional

peaks of intensity originating from AlN- and GaN-rich areas

have been detected.

The resolution function (�qrad, �qang) depends on theangular divergence of the incident and diffracted beams as

well as on the monochromaticity of the X-ray beam. The energy

resolution of the Si(111) crystal monochromator used in our

investigation is �E/E ’ 0.0001, where the energy E is 12 keV.Since the scattering plane corresponds to the vertical plane,

the resolution function in reciprocal space depends on the

vertical divergence of the incident beam, which was defined by

the vertical opening of the slit (0.3 mm) situated upstream

from the sample. The angular resolution (0.004�) of the

diffracted beam was predefined by the channel size of 50 mmand the distance between sample and detector of 700 mm.

Fig. 1(a) (right) is a schematic representation of the non-

coplanar diffraction geometry used for GID measurements,

which were performed at an energy of 8 kev using the

MYTHEN 1K line detector in combination with an Si(111)

crystal analyser. In this case the resolution function (�qrad,�qang) was defined by the crystal analyser resolution and theenergy resolution of the Si(111) crystal monochromator,

which in our investigation is �E/E ’ 0.0001 at an energy E of8 keV.

In the GID geometry the diffracted intensity originates

from the crystalline planes (hkl) perpendicular to the sample

surface. In noncoplanar geometry the incident and the

outgoing wavevectors are each defined by two angles: (�i, �i)and (�f, �f), respectively (see Fig. 1a, right). However theincident angle �i and the outgoing angle �f are relatively smalland close to the critical angle of 0.336�, and will be neglected

in the calculation of the components of the scattering wave-

vector. The angles �i and �f are the angles between the incidentbeam and the outgoing beam, respectively, and the diffracting

planes.

For GID reflections the components of the scattering

wavevector Q in the radial and angular directions are given by,

respectively,

qrad ¼ ð2�=�Þ sin �f þ sin �ið Þ; ð3Þ

qang ¼ ð2�=�Þ cos �f � cos �ið Þ: ð4Þ

For each sample, RSMs of the 10.0 and 11.0 GID reflections

were recorded. Fig. 1(b) (right) gives a typical RSM of the 10.0

reflection. The broadening of the reflection in the angular

direction qang is related to the defect density, while from the

radial direction qrad one can determine the in-plane d spacing.

3. Symmetric X-ray diffraction from the AlGaN epilayersystem

In order to characterize the structural parameters (i.e. crys-

talline parameters, defect densities etc.) of Al0.2Ga0.8N

epilayers, we compared in our previous study (Lazarev et al.,

2012) the RSMs of a GaN sample and an Al0.2Ga0.8N sample

having similar overgrowth layer thicknesses of about 2.5 mm.Differences in peak positions have been detected in the RSMs

of symmetric reflections. One of the aims of this paper is to

identify the origin of the Bragg reflections detected in the

RSMs of the AlGaN structure with respect to the individual

layers of the sample.

The evolution of the X-ray pattern as a function of the

overgrowth thickness was examined by recording the RSMs of

the 00.8 reflection shown in Fig. 1(b) (middle) for the different

samples. The intensity distribution along the CTR was derived

from the RSMs and is displayed in arbitrary units in Fig. 2(a).

In order to clearly differentiate between the curves, each

profile in Fig. 2(a) has been multiplied by a factor of five

relative to the previous one.

In order to identify the origin of the different peaks and to

demonstrate accurately the resulting changes in peak position

and broadening as function of the overgrowth thickness, a

decomposition procedure was systematically applied to all

samples.

Figs. 2(b) and 2(c) contain the intensity distribution along

the CTR as well as the contributing profiles for samples A1

and A7, respectively. It should be emphasized that A1 and A7

correspond to the lowest and highest overgrowth layer

thicknesses, respectively. The choice of Lorentzians as a fitting

function for the individual peaks enables us to achieve a good

fit for the CTR profiles of the samples.

We propose to follow the change in the peak intensities

when the thickness of the overgrowth layer increases. For this

purpose we define h1 = 150 nm as the thickness of the AlGaN

layer below the SiNx interlayer and h as the thickness of the

overgrowth layer, varying from 90 up to 3500 nm (see x2.1). Asis shown in Fig. 2(b) for sample A1, where h = 90 nm is less

than h1, the intensity of peak 3 is higher than that of peak 4,

while in the case of sample A7 (see Fig. 2c), where h = 3500 nm

is greater than h1, the intensity of peak 4 is dominant, because

of the increase of the diffracting volume coming from the

overgrowth layer. As a consequence, the increase of the

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J. Appl. Cryst. (2013). 46 S. Lazarev et al. � Dislocation density reduction in AlGaN epilayers 3 of 8

overgrowth layer thickness h leads to a continuous increase in

the peak 4 intensity (see Fig. 2a). In conclusion, peak 3 could

be attributed to the 150 nm AlGaN interlayer and peak 4 to

the overgrowth layer. The decomposition procedure applied

systematically to all the samples does not reveal any change in

the position of peaks 3 and 4. Since peaks 3 and 4 have

different positions in the CTR cut, one can conclude that the

corresponding vertical lattice parameters of the structure

below and above the SiNx mask are different.

A previous study using spatially resolved scanning electron

microscope cathodoluminescence (Neuschl et al., 2010)

revealed the growth of GaN-rich islands in the openings of the

SiNx interlayer, surrounded by AlN-rich areas. These islands

coalesce to a fairly uniform AlGaN layer, corresponding to the

overgrowth layer in this manuscript. The observed diffraction

peaks 1 and 2 occur close to the values of a pure GaN crys-

talline structure [at qrad(GaN) = 97.87 nm�1], while peak 5 is

closer to the peak of pure AlN [at qrad(AlN) = 100.93 nm�1];

these peaks are therefore presumably attributable to the GaN-

and AlN-rich areas, respectively.

4. Application of Monte Carlo simulation to a two-layersystem

Numerical Monte Carlo simulation was applied successfully

for the determination of the edge (De) and screw (Ds) dislo-

cation densities from the diffuse scattering of GaN samples

having nearly the same growth structure as our AlGaN

samples (Barchuk et al., 2010, 2011). For these samples it has

been demonstrated that the TD densities determined from the

Monte Carlo simulation are comparable to those measured by

the etch pit density method. The simulation of RSMs of GaN

samples was carried out with the assumption that the

diffracted intensity distribution originates only from the GaN

layer above the SiNx layer. The comparison of RSMs of

symmetrical and asymmetrical reflections of AlGaN and GaN

by Lazarev et al. (2012) has revealed different behaviour:

contrary to the case of AlGaN, GaN shows only one peak

along the CTR in the RSM of the 00.8 reflection measured

with the same instrumental function. An attempt to apply the

Monte Carlo method to simulate the RSM by considering only

the AlGaN overgrowth layer was not successful in achieving a

good fit with the measured RSM. For this reason a further

development was implemented into the model to calculate the

RSM of a more complex system consisting of two AlGaN

layers, in order to derive independently and accurately the

density of edge and screw threading dislocations of each

individual AlGaN layer.

Because of their relatively low density we do not take into

account the presence of screw dislocations in the calculation of

the RSMs in GID for either the 150 nm interlayer or the

overgrowth layer. Nevertheless, the simulations of the RSMs

of the symmetric reflection of the overgrowth layer and the

150 nm interlayer are slightly different because of the

different types of interfaces: the 150 nm AlGaN interlayer

does not have a free surface and hence does not reveal any

surface stress relaxation since it forms a top interface with the

AlGaN overgrowth layer and a lower interface with the AlN

nucleation layer. In contrast, the overgrowth layer possesses a

discontinuity at the surface, which gives rise to an additional

nonzero vertical component parallel to the diffraction vector

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Figure 2(a) CTR profiles for samples A1–A7 derived from RSMs of the 00.8reflections. The profiles were shifted vertically in order to make thechanges in peak positions and intensities visible as a function of theovergrowth thicknesses. It is observed that the intensity of peak 3decreases while that of peak 4 increases. (b) The CTR profile of the 00.8reflection and the corresponding decomposed peaks for sample A1,where the overgrowth layer is 90 nm, i.e. lower than the thickness of the150 nm AlGaN interlayer. The intensity of peak 3 is higher than that ofpeak 4. (c) The CTR profile of the 00.8 reflection and the correspondingdecomposed peaks for sample A7, where the overgrowth layer is3500 nm, i.e. higher than the thickness of the 150 nm AlGaN interlayer.Peak 4 is dominant because of the increase of the diffracting volumecoming from the overgrowth layer.

of the symmetric reflections, resulting from the displacement

field created by edge dislocations (Shaibani & Hazzledine,

1981).

Since in the case of the overgrowth layer the edge dislo-

cations play an important role in the general intensity distri-

bution in the RSMs of symmetric reflections, De serves as an

input parameter in the simulations of these RSMs. In contrast,

these symmetric reflections are not affected by De in the case

of the 150 nm AlGaN interlayer.

In our experiment the penetration depth in GID was esti-

mated to be less than 150 nm, which enables us to probe solely

the upper part of the overgrown AlGaN layer for samples A2

through A7 with h > h1. Hence, we could determine the

densities of edge TDs for this thickness. This statement does

not apply to sample A1, where the overgrowth layer was only

90 nm, i.e. smaller than the penetration depth. The study of

this sample will be presented in the next section. A detailed

description of the different steps involved in the application of

Monte Carlo simulation to RSMs was previously introduced

by Barchuk et al. (2010).

To derive the TD density the RSMs of the GID reflections

10.0 and 11.0 were simulated. The simulated RSMs were then

convoluted with a resolution function to achieve a finite

experimental resolution. The resolution function was chosen

as a two-dimensional Gaussian with a width corresponding to

the coherently irradiated area of the sample surface estimated

for our experimental setup. A similar procedure for GaN films

is described by Barchuk et al. (2011). The comparison of

experimental and simulated RSMs for sample A4 is shown in

Figs. 3(b) and 3(c). In the experimental 11.0 RSM (Fig. 3c) one

can see some asymmetry in the radial direction, which indi-

cates the presence of a local difference in the in-plane lattice

parameters. However, for the TD density determination it is

more important to fit the tails of the peaks, and therefore this

asymmetry does not affect the accuracy or reliability of the

values determined by the simulation.

Subsequently, Ds was determined by

considering a system with two AlGaN

layers as described above. Contrary to the

GID method, the penetration depth in the

coplanar diffraction geometry was about

5000 nm for the 00.8 reflection, so all layers

deposited on the sapphire substrate

contribute to the RSM.

In the previous section it was shown in

Fig. 1(b) (middle) that two peaks

appearing in the RSM of the 00.8

symmetric reflection are attributable to the

AlGaN overgrowth layer and to the

150 nm AlGaN interlayer. Consequently,

the intensity distribution of the RSMs

represents the overlap of these two peaks.

The RSMs from these two layers were

simulated independently and then the

intensity distributions were superposed.

The densities of screw dislocations in both

layers were derived after achievement of

the best agreement between experimental

and simulated RSMs (see Fig. 3a). The qangaxis is chosen as the (10.0) direction and

the qrad axis as the (00.1) direction. One

can observe some disagreement between

the experimental and simulated RSMs of

the 00.8 reflection in the lower qrad values

due to the fact that the GaN-rich peak was

not taken into consideration in our simu-

lation model.

In addition to comparing the shapes of

intensity distributions in the RSMs we

analysed their cuts along the angular

directions, following the approach intro-

duced by Kaganer et al. (2005). For the

symmetric 00.8 reflection we compared the

cuts through both peaks. The experimental

and simulated profiles in log–log repre-

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Figure 3Measured (red, dots) and simulated (blue, solid) reciprocal-space maps for the reflections 00.8(a), 10.0 (b) and 11.0 (c) for sample A4. The intensity steps are in logarithmic scale: 100.5 (a) and100.25 (b), (c).

Figure 4The cuts through the main peak of symmetric reflection 00.8 along the angular direction qang; theasymptotic behaviour of cuts obeys q�2ang � q�3ang (a). The cuts through the maximum of intensity ofthe 10.0 GID reflection (b) and the 11.0 reflection (c). For all the samples, the experimentalcurves are depicted by dots and the theoretical curves by solid lines.

sentation are in good agreement, especially in the diffuse part

of the intensity distributions, which confirms that our model

was applied successfully and could be used for accurate

determination of TD densities. The cuts through the main

peaks of symmetric reflections are depicted in Fig. 4(a). The

asymptotic behaviour of the tails seems to follow a q�nang law,

with n between 2 and 3. This indicates the presence of screw

TDs as the prevailing type of dislocation, as is proposed by

Holý et al. (2008). In a similar way, the cuts through the

maximum of intensity of the GID reflections are displayed in

Figs. 4(b) and 4(c). In contrast to the symmetric reflections, the

asymptotic behaviour of GID reflection cuts is mainly influ-

enced by the edge type of TDs. The tails of the curves lie

between q�1ang and q�2ang rather than q

�2ang and q

�3ang, as might be

expected from the universal law q�3ang (Kaganer et al., 2005).

One possible explanation is the excess of the edge TD density

with respect to the screw TD density.

5. Determination of the edge dislocation density belowand above the SiNx nano-mask

Our TEM investigation has led to a comprehensive under-

standing of growth and defect reduction mechanisms occur-

ring by the formation of dislocation half-loops at the SiNxinterface. A qualitative estimation of the dislocation densities

was based on the observation of TEM images (Klein et al.,

2011, 2010). In fact, a homogenous edge-type dislocation

density has been observed in the layer below SiNx, while

defect-free and defect-rich areas have been detected side by

side on a scale of several hundreds of nanometres above the

SiNx nano-mask. In order to determine quantitatively the

efficiency of the SiNx nano-mask in the reduction of threading

dislocation densities, it is very important to probe the AlGaN

layers below and above the SiNx nano-mask simultaneously

with X-rays. For all samples, the GID 11.0 reflection was

measured with a maximum achievable penetration depth of

150 nm, calculated using an approach described by Dosch et al.

(1986). For samples A2–A7 the overgrowth thickness h was

greater than the penetration depth of 150 nm, and only for

sample A1 was it possible to measure simultaneously the

signals from the overgrowth layer completely and those from

the 150 nm AlGaN interlayer partially (60 nm). Consequently

the RSM of the 11.0 GID reflection of sample A1 shows

different features in comparison to the remaining samples. The

RSM of the 11.0 reflection presented in Fig. 5 (left) shows the

presence of two overlapped signals shifted along the qraddirection and originating from the overgrowth layer and the

150 nm interlayer. In order to derive the TD densities from the

overlapped signals, Fig. 5 (middle) illustrates the decomposi-

tion procedure applied to the RSM, involving decomposition

into two-dimensional Lorentzians. The parameters of the

Lorentzians were determined from the decomposition of the

radial and angular scans through the maxima of the signals

indicated in the RSM in Fig. 5 (left). The two derived RSMs

were simulated separately by considering two separate layers

having 90 and 150 nm thicknesses. The RSMs simulated by the

Monte Carlo method for the two layers are compared with the

decomposed RSMs in Fig. 5 (right). The edge dislocation

densities were consequently determined from the best fit and

were found to be 6.4 � 1010 cm�2 for the 90 nm AlGaNovergrowth layer and 9.8 � 1010 cm�2 for the 150 nm AlGaNinterlayer. This method will allow us to quantify the efficiency

of the SiNx nano-mask, as discussed in the next section.

6. Scaling law of the threading dislocation densityreduction

The De and Ds data derived from the Monte Carlo simulation

are plotted in log–log scale as a function of the overgrowth

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Figure 5The 11.0 GID reflection of sample A1 (left), decomposition of the overlapped signals from the 150 nm AlGaN interlayer and the 90 nm overgrowth layer(middle), and Monte Carlo simulations of the peaks (right).

thickness h in Fig. 6(a). The fitting of the De variation with a

scaling law h�n gives an exponent n = 0.95.

We define here the efficiency factor of the SiNx nano-mask

as the ratio between De = 9.8 � 1010 cm�2 of the 150 nmAlGaN interlayer and De of the overgrowth layer thickness h

(Fig. 6b). The increase of the overgrowth layer thickness

conspicuously improves the efficiency of the SiNx nano-mask.

In fact, this factor increases up to 52 for the overgrowth

thickness of 3500 nm.

As described by Klein et al. (2011), the dislocation bending

leads to the annihilation of TDs by the formation of disloca-

tion loops. A growth model was proposed by these authors,

illustrated in Figs. 10 and 11 of Klein et al. (2011), where it is

demonstrated that the overgrowth will favour the coalescence

of two nearby islands leading to small areas with TDs at the

surface. This dislocation reduction mechanism proposed by

the model leads to the enhancement of the efficiency factor

with increasing overgrowth thickness h.

The value of Ds is found to be lower than De for all over-

growth thicknesses h, and follows a scaling law h�n, with an

exponent n = 0.18 for the overgrowth layer and n = 0.33 for the

150 nm AlGaN interlayer. A similar result was derived from

high-resolution TEM investigation (Klein et al., 2010, 2011).

Furthermore, Ds of the overgrowth layer is about a factor of 2

lower than Ds of the 150 nm AlGaN interlayer. This demon-

strates that the SiNx nano-mask plays a role in the reduction of

Ds as well as of De.

7. Conclusions

In this work we found a local difference in the d spacing along

the growth and the lateral directions as determined by the

analysis of symmetric and GID reflections, respectively.

The crystalline lattice parameters of the 150 nm AlGaN

interlayer and of the overgrowth layer are different. Our

investigation of the AlGaN epilayer shows evidence of the

presence of several phases, including two AlGaN phases as

well as GaN- and AlN-rich phases. These latter two have not

been the main focus of this manuscript.

By a systematic study of the intensity distribution along the

CTR of the 00.8 symmetric reflection we were able to attribute

the different peaks to the different individual layers

composing the AlGaN epilayer system. We conclude that the

lower and upper main peaks of the symmetric reflection

originate from the 150 nm AlGaN interlayer and from the

overgrowth layer, respectively. The decomposition of the CTR

profile for all the samples into several peaks shows a change in

the peak intensities but no change in the peak positions, which

means that the local difference in the d spacing is not influ-

enced by the overgrowth thickness.

Our simulation based on the Monte Carlo method has

proven to be reliable in modelling the RSMs in coplanar and

noncoplanar geometries of a two-layer system. The experi-

mental and the Monte Carlo-simulated RSMs are in good

agreement. This allows us to derive screw and edge-type

threading dislocation densities with an error of less than 15%.

Furthermore, the reduction of De follows a scaling law h�n

with the overgrowth thickness, where n = 0.95. The asymptotic

behaviour of the tails of the symmetric reflections obeys a q�nanglaw, with 2 < n < 3, while the tails of the diffraction peaks in the

GID follow a q�nang law, with 1 < n < 2. The deviation of

asymptotic decay from the universal law q�3ang, particularly in

GID, is due to the large edge-type TD densities.

We have demonstrated that the analysis of diffuse scattering

by using the simulation of decomposed RSMs of GID reflec-

tions is a powerful method to determine separately the TDs in

a two-layer system, e.g. below and above an SiNx nano-mask.

We would like to thank Dr Gernot Buth, beamline scientist

at the SCD beamline at ANKA (Karlsruhe Institute of

Technology), for his support during beamtime and Dr Stephen

Doyle for his help with the manuscript. The epitaxial growth

of the AlGaN layers under investigation was financially

supported by the BMBF.

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Figure 6(a) The variation of Ds determined from coplanar high-resolution X-raydiffraction and De determined from GID measurement as a function ofthe overgrowth thickness h. (b) The variation of the efficiency of the SiNxnano-mask in dislocation reduction with overgrowth thickness h.

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