G46

- XRD Master HR -

A new X-ray diffractometer for the determination of the metric of

single crystals and epitaxial layers using the extended Bond technique

N. Herres1, P. Koidl1, R. Stabenow2 and A. Haase2

1 Fraunhofer Institut fr Angewandte Festkrperphysik (IAF), Tullastrasse 72,D-79108 Freiburg, Germany. Email: herres@iaf.fhg.de

2 Rich. Seifert & Co., Bogenstrasse 41, D-22926 Ahrensburg, GermanyEmail: r.stabenow@roentgenseifert.de

Motivation

High-resolution X-ray diffraction (HRXRD) is an established tool for the structural characterisationof crystalline substrates and thin films. Lattice parameters, superlattice periods, layer thicknessesand the structural parameters deduced from these can be determined highly reliable and non-destructively. Materials development in the microelectronics industry relies to a great deal on thedata provided by HRXRD techniques.

Heteroepitaxial layer systems for which materials problems have been solved, rarely pose anyproblems to the industrial diffractionist. In daily life, it is when technologists encounter problemsduring growth or processing, that the X-ray diffraction laboratory is challenged. In these cases, theX-ray investigation very often is hampered by unwanted and unintended effects like samplecurvature, tilt of the layer relative to a substrate lattice oriented off-axis, subgrain tilts, partialrelaxation etc. Here, we present a new type of X-ray diffractometer which helps to overcome many,if not most, of these problems by virtue of its goniometer design and measurement procedures.

Diffractometer set-up

The new X-ray diffractometer was designed to accomodate, adjust and measure samples with sizesfrom below 1 mm2 up to 150 mm wafers supplied in cleanroom compatible trays ( 200 mmdiameter, 15 mm in thickness). Simple rocking curve scans and automated measurementsequences involving all goniometer axes can be performed.

The conditioning of the primary X-ray beam is done with a parabolically bent graded multilayermirror and two symmetrical channel-cut Ge-220 crystals (Bartels-DuMond type). A discriminationof the diffracted X-ray beam, if required, may be done with either slits or a second Bartels-DuMondtype analyzer.

Six goniometer axes (, , , x, y, z) serve to adjust the sample in a well-defined way prior to themeasurement(s). Two axes (, 2) position and measure under closed-loop angular control usingaccurate incremental optical encoders (Fig.1).

2Fig. 1 Scheme of the new X-ray goniometer

Fig. 2 The X-ray diffractometer XRD Master HR at IAF Freiburg

2

x

z y

Axis

xyz

2

Range

75 mm 75 mm20 mm

-175 ... +170-20 ... +20-360 ... +360

-10 ... +190-117 ... +157

(-137 ... +137)

3Principle of operation: the extended Bond technique

The Bond method (Bond 1960) is an established technique for the determination of the absolutelattice parameter of bulk single crystals to an accuracy of the order of a few ppm. Using a singlereflection, the angle between two crystal positions at which the Bragg condition is fulfilled, serves tocalculate the interplanar spacing of the reflecting lattice planes. For cubic crystals withoutinhomogeneous strains, the measurement procedure is quick and accurate. If the unit cell has lowersymmetry, a re-setting of the crystal and additional measurements for each lattice parameter to bedetermined are required (Grosswig 1983).

A measurement sequence termed the extended Bond method (Grosswig 1985) overcomes thedisadvange of the Bond method by measuring not one but three complanar lattice parameters (twobasic vectors and the angle inclosed between them) of an unstrained or strained unit cell in onestate of adjustment of goniometer and sample. This technique has been developed to determinehighly precise lattice parameters with non-cubic bulk crystals like quartz monocrystals. Itsapplication for precision measurements of strained layers has not become popular becausecommercial high-resolution diffractometers up till now did not to meet two important experimentalrequirements: first, a zone-axis of the crystal has to be adjusted parallel to the -axis prior to themeasurement to better than 0.05. Second, angular measurements (-axis) with an accuracy of theorder of 0.001 must be possible over an angular range () of at least 150.

A measurement run comprises diffraction from several (hkl) lattice planes belonging to the samecrystallographic zone. Within the measuring procedure no re-adjustment between measurements isacceptable. Thus, a crystallographic zone axis has to be aligned parallel to the direction of themeasurement axis of the goniometer. (Fig. 3)

Fig. 3 The angles between several (hkl) planes of a crystallographic zone can bedetermined from diffraction peak positions in -scans around the zone axis. The useof symmetrically equivalent lattice planes (hkl)a+ and (hkl)a- relative to the surface(hkl)0 facilitates the determination of lattice parameters.

Diffraction data of film and substrate are evaluated independently. Thus, e.g. a films peak positionsin all reflection ranges of a crystallographic zone are combined to determine the films latticeparameters within the plane of dispersion (= perpendicular to the zone axis). A minimum of 4reflection ranges (2 symmetrical and 2 asymmetrical) is usually sufficient to obtain reasonablyaccurate data (Fig. 4). The use of pairs of "asymmetric" lattice planes (hkl)a with the same d-spacingand the same non-zero inclination relative to the surface (hkl)0 facilitates evaluation. A correctionfor refraction is necessary, especially, if shallow incidence angles are used.

2

zone axis

incidentX-ray beam

2

2 = (hkl)a+ - (hkl)a-

.

.

. .

.

(hkl)a+1

a -1(hkl)

a -2

(hkl)

(hkl)a+2

(hkl)s0

zone axis

22 21

4 Fig. 4 Symmetrical and asymmetrical back reflection geometries used with the extendedBond technique. From symmetrical diffraction ranges hkls the spacing of latticeplanes parallel to the surface (and the zero-point error of the reading) is obtained.This in mind, the translation period parallel to the surface follows indirectly fromthe angle between metrically equivalent lattice planes determined from peakpositions measured with shallow (hkla-) and steep (hkla+) incidence.

If higher accuracy of the lattice parameters is required, more reflection ranges can be added to theminimum of 2 symmetrical and 2 asymmetrical reflection ranges. In this case, a least squaresminimization of the deviations between observed and calculated reflection positions is useful andmay conveniently be performed with Excel sheets. This has the added advantange, that it facilitatesfurther processing of the data to calculate an unstrained metric and (if needed) the composition.

Clearly, an uniaxially or biaxially strained unit cell cannot sensibly be refined using a cubic metric.Thus one has to consider beforehand, which kind of distortion is applicable.

Heteroepitaxial films deposited on exactly oriented cubic (100), cubic (111) and hexagonal (0001)surfaces usually experience isotropic stresses and strains within the interface to the substrate. Thisleads to a distortion of the films unit cell which becomes either tetragonal, rhombohedral orhexagonal. The direction of compression or elongation of these unit cells is perpendicular to thefilms surface along the unique axis of the crystal system. The resulting strains within the layer arerelated to each other via expressions given in Table I (Hornstra 1978, Seegmller 1988, Van deWalle 1989).

In symmetrical back reflection geometry the lattice spacing parallel to the unique axis (e.g. "c" in thecase of a tetragonal or hexagonal unit cell) is measured. The peak positions of the film obtained inasymmetric reflection geometry using shallow and steep incidence (Fig. 4), enable the c/a ratio to becalculated. This is because the angle between the surface and a lattice plane inclined to the surfacedeviates with increasing distortion of the unit cell from its undistorted value (Fig. 5).

For the three cases under consideration, Table II gives examples for possible combinations ofsymmetric and asymmetric reflections. In addition a formulation of the angle between surface(hkl)0 and reflecting asymmetric lattice plane (hkl)

a is given a function of the c/a ratio.

incidentX-ray beam

incidentX-ray beam

+2

S+ = Bragg + zero

hkls+0

+2

hkla+

a+ = Bragg+ + zero

incidentX-ray beam

incidentX-ray beam

2

S- = 180 Bragg + zerohkls-0

+2

hkla-

a- = Bragg + zero

5Table I

Lattice plane spacing perpendicular to the films surface (e.g. d00L): unstrained: d strained: d

Lattice plane spacing parallel to the films surface: (e.g. dHK0): unstrained: d=strained: d=

Strain definitions and relationship:

= d / do -1 = strain perpendicular to the film surface = = d= / d=o 1 = strain parallel to the film surface

= - = Dhkl = fundamental relationship for lattices straineduniaxially along a highly symmetric direction

The elastic parameter Dhkl depends on a films elastic properties (material and orientation):

D001 = 2 c12 / c11 for a cubic film uniaxially strained along

D111 = 2 (c11+2c12-2c44) / (c11+2c12+4c44) for a cubic film uniaxially strained along

D0001 = 2 c13 / c33 for a hexagonal film uniaxially strained along

These uniaxially strained unit cells can be mathematically unstrained using the following equation:

d = ( d + d= Dhkl Thkl ) / ( 1 + Dhkl ) with Thkl = d / d= serving as a normalization factor[i.e.: T001 = d001 / d100 = 1 ; T111 = d111 / d110 = 3 / 2 ; T0001 = d00.1 / 2d11.0 = c0 / a0 ]

Table II

Unit cell distortion Surface Zone axis X-ray reflections symmetrical asymmetrical

Angle between a lattice plane (hkl) and the crystal surface (001) or (00.1):

Lattice parameter ratio c / a given as a function of the angle :

1cos

1: 22

2

22

2 +

+=

ac

lkhtetragonal

hkl1

34

cos1: 2

2

2

22

.2 +

++=

ac

lhkkhhexagonal

lhk

+

= 22

2

2 1cos1:

khl

actetragonal

hkl 431

cos1: 22

2

.2

++

=

hkkhl

achexagonal

lhk

cubic tetragonal (001)

(001)

00l (e.g. 004) hhl (e.g. 335, 335)

00l (e.g. 004) h0l (e.g. 206, 206)

cubic rhombohedral (111)

(0001)

hexag. indexed:

hhh (e.g. 333) hhl (e.g. 117, 551)

h0.l (e.g. 6069, 6069)00.l (e.g. 00.9)

hexagonal hexagonal (0001)

h0.l (e.g. 1015, 0115)00.l (e.g. 00.6)

00.l (e.g. 00.6) hh.l (e.g. 1123, 1123)

6Fig. 5 The state of strain of a film (shown here for a cubic film on a cubic (001) substrate)can be deduced from the angle between (sym-)metrically equivalent lattice planes.

The determination of a strained metric is accomplished from peak positions of a film alonewithout recurrence to eventually unreliable or unavailable lattice parameters of a substrate.

Whatever the state of strain of the film is, this routine will return values for the lattice periodsperpendicular and parallel to the films surface. But, how reliable are they ?

A routine check is to re-calculate the substrates lattice parameters from its reflection positions, too.This should normally return substrate lattice parameters close to the unstrained value(s). Comparingthe strains of the substrate with that of the film, the substrate strains will normally be of oppositesign compared with the film strains.

But what happens, if the relaxation mechanism leads to an inhomogeneous strain situation at theinterface? Crystals with sphalerite structure often relax "using" dislocation movement on thesymmetrically (and energetically) non-equivalent {111}A and {111}B planes. In this case, theresulting non-isotropically strained unit cell is unlikely to retain its high symmetry. Lowering of thesymmetry and tilting of the layer with respect to the substrate has been observed (Grundmann1990). Here, a rotation around the nominal surface normal (i.e. a rotation around with no need forre-adjustment), followed by a new measurement sequence using a different zone axis, will shed lightonto the situation. If present, this procedure with unambiguously resolve the orthorhombic,monoclinic or even triclinic metric.

And how does the extended Bond technique compare to state-of-the-art diffraction space mapping ?In fact, measuring the peak position of a partially or fully relaxed peak is by at least a factor of 20quicker using an scan than it is measuring even only a single diffraction space map. With theextended Bond technique four such scans are needed and then free completely from ambiguitiesdue to tilt, zero-point shifts and eventually strained substrate lattice parameters which cancomplicate the evaluation of a diffraction space map and then require measurement and evaluationof additional diffraction space maps.

In order to check the technique we performed both, diffraction space maps and extended Bondmeasurements, on several samples: a tetragonally strained (and partially relaxed) InGaAs film onGaAs(001), a tetragonally strained InAs/GaSb-superlattice on GaAs(001) (Herres 1996), arhombohedrally strained PbSrTe/PbTe-superlattice on BaF2(111) and a hexagonally strainedInGaN/GaN heterostructure on Al2O3(0001). It is outside the scope of this paper to present thesecomparisons. We found in all cases, that the lattice parameter values which were determined usingthe extended Bond technique were at least as accurate as those obtained using diffraction spacemaps.

The following part of this paper will give a brief account of the goniometers capabilities. After this,the lattice parameter determination of a 2 m thick layer of GaN deposited on sapphire (0001) willbe demonstrated. A recent investigation of some 30 nm thick ternary layers of InxGa1-xN depositedon a GaN buffer on sapphire (0001) will be used as an example on how to unstrain a strainedmetric (mathematically and in a graphic manner) in order to determine a layers composition.

substrate

dfilm

dsubstrate

dfilm = d=film

dsubstrate = d=substrate

d film,strained d= film,strained

dsubstrate d=substrate

d=film = d=substrate strained

strained unstrained

completely unstrained pseudomorphically strained

unstrained

film

7Test of the goniometers angular accuracy

The key to precision is the reduction of unsystematic errors by multiple measurements. The key toaccuracy lies in the knowledge and elimination of systematic errors. In order to check theperformance of the goniometer in these respects, extended Bond measurements with a zone-refinedsilicon (111) sample of dimensions 15 mm x 10 mm x 2 mm were executed.

Symmetrical reflections:hkl 2Bragg (+2) (-2)111 0.0 28.446 14.223 165.577222 0.0 58.853 29.429 150.571333 0.0 94.953 47.476 132.524444 0.0 158.629 79.315 (100.685)

Asymmetrical reflections: hkl 2Bragg (-) (+)335 14.420 136.893 54.026 (97.134)224 19.471 88.031 24.545 116.514331 22.002 76.378 163.812 60.190440 35.264 106.709 161.909 88.618115 38.942 94.955 8.537 93.581444 70.529 158.631 (171.211) 149.843

Fig. 6 Stereographic projection of the intersecting zones of the silicon(111) specimen.The calculations used an X-ray wavelength of = 154.060 pm, a silicon lattice parameterof aSi = 543.109 pm (T = 23 C, p = 1 bar) and a refraction parameter = 7.58 x 10-6.

Prior to the measurements, the sample has been adjusted on and circles using the symmetrical333 reflection until the (111) lattice plane was aligned perpendicular to the axis. Using theasymmetric 115 reflection (in shallow incidence), the position of a zone axis was thenoptimized by rotation around the axis, until it was set parallel to the axis.

The measurement procedure comprised the scanning of 17 reflecting positions along the threeintersecting zones shown in Fig. 6. After completion of all 17 ranges in one zone, the samplewas rotated on by a 60 angle and measurements in the neighboring zone were started. A total of17 x 6 = 102 scans were done in a first run. In order to test the diffractometers stability and thereproducibility of peak positions this run has been repeated on the following day.

All peaks showed Gaussian peak shapes with peak widths expected for the Ge220 Bartels-DuMondsetup-up: FWHM between 0.0029 (symmetrical 111 reflection) and 0.031 (444 reflection at shallowincidence). The reflection positions () obtained from peak fitting were compared with calculatedreflection positions based on the literature value of silicon (Windisch 1990), taking into account theambient temperature and the refraction correction. The actual value for the X-ray wavelength used(CuK1) was refined using the measured silicon 333 and 444 reflections and amounted to =(154.0622 0.0018) pm.

After this, the measured and calculated peak positions were compared. The average of thedifferences between calculated and measured peak positions equals the zero-point shift zero. Thisshift is due to misalignement of the sample and eccenticity of the axis and was below 0.009 withall 12 runs. Accounting for this zero-point shift is automatically done, if one uses the Bondtechnique. In our case, it is useful to substract the zero-point shift because the remaining differencesbetween measured and calculated peak positions are solely due to angular reading errors (Fig. 7).

The reproducibility of a single reading can be expressed by the standard deviation and amountedon average to 0.0012 0.0004 (1) for the 12 runs measured. The accuracy of lattice parametermeasurements will mainly depend on a proper calibration of the wavelength and the choice ofreflections (prefering large Bragg angles whenever possible).

444115

224

335

[001] 440

331

[100]

[010]

[101] [011]

[110]

444333222111

8Fig. 7 Measured peak positions () and zero-shift corrected angular differences() relative to the calculated silicon peaks on three zones

If one uses the Bond technique with only a single reflection at the relatively low angle 2 = 90 theaccuracy of the wavelength plus lattice parameter determination is not likely to exceed 2x(0.0012/tan 45) 4 x 10-5, a value that will still be acceptable with most thin films. Using the extendedBond technique with several reflections and large Bragg angles, an accuracy better than 10-5 ispossible even with thin films (Gehrsitz 1999).

Measurement example I: 2.7 m thick GaN buffer layer on sapphire (0001) substrate

The unit cell of hexagonal binary nitride films (AlN, GaN) deposited onto (0001) oriented sapphirewafers by MOCVD is usually elongated along the [0001] growth direction (Fig.8). This is due tostresses induced by thermal expansion differences between the film and the sapphire substrate oncooling down from deposition temperatures around 1000C.

Fig. 8 Thermal expansion of sapphire and hexagonal nitrides (GaN, AlN, InN)within the growth plane and resulting uniaxial deformation of the unit cellof a nitride film grown with (0001) onto a sapphire (0001) surface.

10100

1k10k

100k1M

10M

331

222

111

111

222

333

444

333

444

224

331

224

115

335

440

115

440

X-ra

y in

tens

ity [c

ps]

0 20 40 60 80 100 120 140 160 180

-0.004-0.0020.0000.0020.004

standard deviation (1) = 0.0012

diffe

renc

e

[]

[]

0 200 400 600 800 10000

2

4

6

8sapphire InN GaN AlN

rela

tive

ther

mal

exp

ansi

on

a/a

[x10

-3]

temperature T [C]

c

a1a2

a3

9The epitaxial relationships observed between film and substrate are based primarily on symmetry,not on metric coincidences, thus we cannot expect to find reflections of the sapphire substrate inthe vicinity of nitride film reflections. As the extended Bond technique does not require substratepeaks to determine a layers metric, it is well suited to determine strains in these nitride films. Theexperimental procedure to measure the metric of a 2 m thick GaN buffer deposited on a (0001)sapphire substrate is as follows:

The sample is fixed with rubber cement to a plexiglas sample holder in the center of the goniometer.To speed things up, the 00.12 reflections of sapphire are used in order to align the [0001] directionof the substrate parallel to the axis. Then the 10.10 sapphire reflection is used to bring a zone axis of sapphire parallel to the axis by a rotation around . As we know the epitaxialrelationship, an additional 90 rotation around now brings a zone axis of GaN parallel tothe axis. The whole alignment procedure should not take longer than say 20 minutes. With thisaccomplished, the sample is ready for automated measurement of all GaN reflections belonging tothe zone (i.e. 10.5, 10.6, 20.5, 20.4, 00.2, 00.4, 00.6).

Table III List of observed, corrected and re-calculated angles of GaN reflectionsfrom a strained GaN film on (0001) sapphire

Table III shows experimentally determined peak positions from such a run. The 00.6 and 00.4 peaksare used to calculate the lattice parameter c and may be used to do a zero-point correction (notnecessary but convenient). The asymmetric reflections are used to deduce the c/a ratio from theangular difference between reflecting positions with shallow and steep incidence. Using the Excelsolver option, the refinement is straight forward. In the present case the following strained latticeparameters were obtained: a = (318.624 0.015) pm, c = (518.774 0.014) pm.

Finite element modelling for this 2.7 m GaN film deposited at 1000C on a 500 m thick 2-inchdiameter (0001) sapphire substrate calculated a (compressive) film stress parallel to the growthsurface of (0001) = - 0.79 GPa, an in-plane film strain of (0001) = - 0.17 % and a substrate curvatureradius of R = 7.0 m. This compares with the "in-plane" strain (0001) = - (0.12 0.05) % of the GaNfilm determined using the extended Bond-technique and the curvature radius R = (6.86 0.05) mof the sapphire wafer measured by spatially resolved X-ray diffractometry.

reflection incidence [deg.] [deg.] [deg.] [deg.]experiment 0-corrected calculated difference

00.6 symm ( -2 ) 117,000000.6 symm ( +2 ) 62,9826 62,9913 62,9901 0,001200.4 symm ( -2 ) 143,552800.4 symm ( +2 ) 36,4286 36,4379 36,4391 -0,001210.5 shallow 31,8685 31,8788 31,8791 -0,000410.5 steep 73,0806 73,0909 73,0913 -0,000420.5 shallow 31,3194 31,3297 31,3288 0,000820.5 steep 105,2049 105,2152 105,2146 0,000520.4 shallow 11,3684 11,3787 11,3799 -0,001320.4 steep 97,8222 97,8325 97,8334 -0,000910.6 shallow 51,6028 51,6131 51,6119 0,001210.6 steep 86,3979 86,4082 86,4070 0,0012

10.5 20,6061 20,6061 0,0000 20.5 36,9428 36,9426 0,0002 20.4 43,2269 43,2267 0,0002 10.6 17,3976 17,3976 0,0000

10

Measurement example II: 30 nm InxGa1-xN film on a thick GaN buffer layer on sapphire (0001)

(AlGaIn)-Nitrides are important new materials for photonic and electronic applications.Composition and strain of the InxGa1-xN and AlxGa1-xN layers have to be tailored precisely. Tailoringthe extended Bond technique not for accuracy but for speed, we can restrict the number of reflectionranges to the minimum of 4. This has been done with a series of samples consisting of a 30 nm thinInxGa1-xN film on top of a thick GaN buffer layer on a (0001) sapphire substrate. (Wagner 1999)

The procedure to align the sample inside the diffractometer is the same as before. Fig. 9 showsresults of the measurement sequence: the symmetric 00.6 reflection is measured twice (in the 2and +2 range) and a pair of asymmetric reflections (e.g. 10.5) is measured at shallow and steepincidence. All measurements are done in a single run and (in this case) yield the strained latticeparameters of the GaN buffer layer and the InxGa1-xN film.

Fig. 9 Sequence of four measurements used for lattice parameter measurementsaccording to the extended Bond technique. Peaks are fitted with Gaussians

The strained metric is deduced from the measurement of the Bragg angle and the angle betweensymmetricaly equivalent (10.5) lattice planes (using the formula given in Table II). Metrices of theInxGa1-xN film and the GaN buffer layer are determined independently. With knowledge of thestrained metric, a determination of the composition (Indium-content: x) of the ternary InGaN layercan be accomplished by solving a cubic equation for x (Schuster 1999).

The approach used here is similar but a little bit more transparent: in a first step the unit cell isunstrained mathematically using the formula given in Table I (for this we need to know the elasticparameter Dhkl and a normalization factor Thkl). The composition is then determined by interpolationalong a calibration curve (in the simplest case: a linear interpolation between structural parametersof the binary end-members). Usually, the elastic data depend somewhat on the composition, thus aquickly converging recursion procedure as shown in Fig. 10 is necessary to obtain the composition.

This procedure works, no matter if a film is pseudomorphically strained, partially relaxed, orcompletely relaxed, as long as the strain can be described by one of the three uniaxial distortionsgiven in Table II. The determination of the unstrained lattice parameters relies to some extend onexternal input data for the materials parameters: the elastic data Dhkl and (in the hexagonal case) thefactor Thkl. The accuracy of the composition strongly depends on the calibration curve used totranslate the unit cell parameters into chemical content.

117 118 119

GaN

InGaN

00.6 (-2)

X-ra

y in

tens

ity (l

og. s

cale

/ ar

b. u

nits

)

61 62 63

InGaN

GaN

00.6 (+2)

72.0 72.5 73.0

asymmetric reflections10.5 c/a

symmetric reflections00.6 c

10.5 (+2, + )steep incidence

GaN

InGaN

incidence angle [deg.]30 31 32

GaN

InGaN

10.5 (+2, - )shallow incidence

11

Fig. 10 Evaluation procedure used to determine the composition of auniaxially strained hexagonal InxGa1-xN film.

An exact graphical representation of the evaluation procedure is given in Fig. 11. Assuming alinear change of the c/a ratio between the end-members GaN and InN, an unstrained unit cell ofInxGa1-xN will be represented by a point along the green line connecting the GaN spot at c = 518.55pm and a = 318.91 pm (Balkas 1995) with the InN spot at c = 570.34 pm and a = 353.83 pm (IAF1998). Straining a GaN unit cell shifts the spot along the dashed red line whose slope is given by theelastic parameter D0001 = 0.42 for GaN (IAF 1999).

Fig. 11 A graphical evaluation procedure to unstrain strained metricesand obtain the InGaN layer composition

Assuming that the slope of these dashed red lines changes linearily with composition up to the InN-value of D0001 = 0.82 (Wright 1997) lines with appropriate slopes can be drawn at 5%, 10% and 15%In-content. These lines serve as guide to the eye when trying to unstrain the InxGa1-xN filmsgraphically. Points representing the experimentally determined strained lattice parameters of severalthin InxGa1-xN films of different compositions and degree of reaxation are given with their error bars.Following the dotted arrows until the green line is reached, yields the unstrained lattice parameters,and by interpolation along the green line also the Indium-content of the InxGa1-xN film.

measurement of peak positionscalculation of strained lattice parameters1

(linear) interpolation of material parameters - elastic parameter: D00.1 = 2c13/c33 - lattice parameter ratio: T = cunstrained / aunstrained

2

3mathematical relaxation: cunstrained = (cstrained + astrained . D00.1 . T) / (1 + D00.1)

4 calculation of the composition (Vegards rule): xIn = (cInGaN,unstrained - cGaN ) / ( cInN - cGaN)

12

Conclusions

The extended Bond technique is an extremely useful tool to determine the strain situation of singlecrystals and heteroepitaxial layers and superlattices. The determination of a strained film metric isaccomplished from the peak positions of the film alone (substrate reflections are not required).

Developed from a conventional HRXRD machine, the new diffractometer "XRD Master HR" meetsthe requirements for routine extended Bond measurements on single crystals and state-of-the-artheterostructures. Six goniometer axes (, , , x, y, z) serve to adjust the sample prior to themeasurement(s), while two axes (, 2) position and measure under closed-loop angular controlusing accurate incremental optical encoders.

The strong point of this diffractometer is its ability to cope with problematic samples. Conventionalrocking curve and diffraction space map measurements profit from a very useful feature of thegoniometer: after suitable -rotation around the diffraction vector, in most cases even samples witha strong potato-chip-like curvature will show good diffraction profiles.

Using the extended Bond technique, the diffractometer is especially suited to characterize thepartially relaxed heterostructures often encountered at the beginning or during the optimization ofdeposition processes. The determination of a strained metric is accomplished from the peakpositions of the film alone without recurrence to eventually unreliable or unavailable peaks of asubstrate. No matter if a films lattice parameter parallel to the substrate is unknown or in doubt, aroutine determination of strain and composition is quick, highly reproducible and nearlyundisputable.

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elastic constants and the lattice parameter of AlxGa1-xAs. Phys. Rev. B 60 (1999) 11601-11610.Grosswig, S.; Hrtwig, J.; Alter, U.; Christoph, A.: Precision lattice parameter determination of coloured quartz

monocrystals. Cryst. Res. Technol. 18 (1983) 501-511.Grosswig, S.; Jckel, K.-H.; Kittner, R.; Dietrich, B.; Schellenberger, U.: Determination of the complanar

geometric lattice parameters of monocrystals with high precision. Cryst. Res. Technol. 20 (1985) 1093-1100.Grundmann, M.; Lienert, U.; Christen, J.; Bimberg, D.: Dependence of structural and optical properties of

In0.23Ga0.77As/GaAs quantum wells on misfit dislocations: Different critical thickness for dislocationgeneration and degradation of optical properties. J. Vac. Sci. Technol. B 8 (1990) 751-757.

Herres, N.; Fuchs, F.; Schmitz, K. M.; Pavlov, K. M.; Wagner, J.; Ralston, J. D.; Koidl, P.; Gadaleta, C.;Scamarcio, G.: Effect of interfacial bonding on the structural and vibrational properties of InAs/GaSbsuperlattices. Phys. Rev. B 53 (1996) 15688-15705.

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