IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 42 (2009) 245402 (7pp) doi:10.1088/0022-3727/42/24/245402

Strain analysis on freestandinggermanium nanocrystalsP K Giri

Department of Physics and Centre for Nanotechnology, Indian Institute of Technology,Guwahati 781039, India

E-mail: giri@iitg.ernet.in

Received 7 July 2009, in final form 12 October 2009Published 26 November 2009Online at stacks.iop.org/JPhysD/42/245402

AbstractWe report on a detailed study of strain in freestanding Ge nanocrystals (NCs) by using x-raydiffraction (XRD) line profile analysis supported by high resolution transmission microscopy(HRTEM) imaging. Freestanding Ge NCs down to ∼7 nm size are synthesized by using theball milling technique and investigated regarding the nature of strain. Detailed analysis of sizeand lattice strain in the NCs reveals that strain is anisotropic in the NCs. NC size and strainanisotropy factor are calculated by taking into consideration a dislocation contrast factor. Theanalysis further suggests that screw type dislocations are the main contributors to the strainanisotropy and the dislocation density and corresponding strain vary with crystallite size, witha maximum of both quantities for NCs produced after 20 h of milling. Direct evidence forstrain caused by dislocations in individual NCs is provided from HRTEM imaging. Relaxationof strain is studied by differential scanning calorimetry, which shows a low temperature heatrelease at ∼310 ◦C, clearly indicating a kind of structural relaxation of the strained NCs. Themethodology presented here is applicable for embedded as well as freestanding NCs of othermaterials. Implications of strain on the optical properties of Ge NCs are discussed.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Owing to their technological importance, the growth andproperties of Ge nanocrystals (NCs) embedded in amorphousSiO2 matrices have been studied by several groups [1].However, the NC size and embedding matrix are knownto strongly influence the structural, thermal and opticalproperties of the embedded Ge NCs [2, 3]. For example,ion-beam-synthesized Ge NCs embedded in an amorphoussilica matrix exhibit large compressive stress in the as-grownstate [4, 5]. Similarly, the presence of compressive strainhas been reported for embedded Ge NCs grown by co-sputtering [6]. Nonuniform distribution of elastic field at theGe/SiO2 interface has been reported [7]. Some of the studieshave particularly focused on the stress evolution and stressrelaxation in Ge NCs [4, 8]. Several studies have attempted tounderstand the structural aspects of Ge NCs that are differentfrom bulk Ge [9]. In order to understand and exploit theintrinsic properties of Ge NCs, it is important to study Ge

NCs that are freestanding or isolated. There have been afew attempts to study the structural and optical propertiesof freestanding Ge NCs grown by selective etching of theembedding matrix [3, 8, 10]. Strain in NCs has been assessedcommonly by Raman spectral features, where both the phononconfinement effect and the stress effect are coupled in thespectra. Zheng et al [3] have studied stress tuning of Ge NCsby Raman spectral analysis.

Ball milling is a powerful technique for large-scaleproduction of freestanding NCs [11] and other nanostructures[12]. However, very little is known about the microstructureof such freestanding Ge NCs. Broadening in the x-raydiffraction (XRD) line profile is usually used to calculatecrystallite size and strain in NCs. However, for crystalswith the anisotropic structure this method must be used withcare. Ungar et al [13, 14] proposed a powerful method ofXRD line profile analysis that takes into account anisotropyin strain and it enables precise determination of particle size,strain and mechanism of strain in NCs. This method has

0022-3727/09/245402+07$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK

J. Phys. D: Appl. Phys. 42 (2009) 245402 P K Giri

been applied successfully to study varieties of nanomaterials[15, 16]. However, to our knowledge, no studies have beenreported on the strain anisotropy of Ge NCs.

In this report, we present detailed investigations on sizedependent strain in freestanding Ge NCs down to ∼ 7 nmsize synthesized by the ball milling technique. Structure andmorphology of the Ge NCs are studied by transmission electronmicroscopy (TEM) and atomic force microscopy (AFM),respectively. Analysis based on the XRD line profile showsanisotropy of strain in the Ge NCs, and direct evidence of strainin individual NCs is confirmed from high resolution TEM (highresolution transmission microscopy—HRTEM) imaging. Wefurther discuss the strain relaxation of Ge NCs at an elevatedtemperature.

2. Experimental details

The freestanding Ge NCs are prepared from high purity(99.999%) Ge powder (Sigma Aldrich) with initial particlesizes < 150 µm by mechanical milling in a planetary ballmill for a duration up to 40 h in a zirconium oxide vialfilled with small balls of zirconium oxide. This ensures thatno metallic contaminants are introduced during the millingprocess. The milling was performed under normal atmosphericconditions, except for 40 h milling that was performed in aliquid environment to achieve ultralow grain size. The powderto ball ratio was taken as 1 : 20 and the ball mill was operatedat 350 rpm. The nanopowder samples obtained after every 5 hof milling were studied by high resolution XRD measurements(Bruker, Advance D8) for the careful determination of averagecrystallite size, internal strain and dislocation density usingthe Ungar et al method [13]. An AFM (Agilent, SPM 5500))was used to study the size distribution and morphology ofthe milled NCs. A 200 KV HRTEM (JEOL-2010) was usedto study the nanocrystallite size and the microstructure ofthe freestanding NCs. DSC measurements were carried outusing a commercial calorimeter (Perkin-Elmer, DSC-7) with aheating rate of 5 ◦C min−1. For the convenience of subsequentdiscussion, we denote the unmilled, 5 h, 10 h, 20 h and 30 hmilled Ge NCs samples as Ge-0, Ge-5, Ge-10, Ge-20 andGe-30, respectively.

3. Results and discussion

3.1. AFM and TEM studies

Gradual reduction in particle size as a result of ball millingwas monitored after every 5 h of milling. Figures 1(a)-(f )show typical AFM and TEM images of the freestanding GeNCs in Ge-5, Ge-20, Ge-30 samples. Figure 1(a) shows theAFM image of the isolated Ge NCs in Ge-5 and the insetshows the histogram of the size distribution obtained from thisAFM. The average size is found to be 38.8 nm for the Ge-5sample. Figure 1(b) shows the TEM image of NCs in Ge-5.Similarly, the average size of the NCs in Ge-20 is found tobe about ∼14.0 nm from the AFM image of figures 1(c) and(d). Figure 1(e) shows the TEM image of NCs in Ge-20 andfigure 1(f ) for that of Ge-30. The inset of figure 1(e) shows a

HRTEM lattice image of an isolated Ge NC of diameter 10 nm.Both the AFM and TEM images reveal that many of the NCs areof nonspherical shape. It is due to the high speed grinding thatthe initial spherical particles break into nonspherical shapesand strain is developed in the milled NCs due to deformation.

3.2. XRD line profile analysis and strain anisotropy factor

The XRD patterns for Ge nanopowders synthesized for variousmilling times are shown in figure 2, which clearly revealgradual broadening of the line shape due to the reductionin the average crystallite size and development of strain inthe NCs. Full width at half maxima (FWHM) of the XRDpattern was calculated by fitting a Lorentzian line shape to theexperimental data. According to the Williamson–Hall (WH)method [17], the individual contribution to the broadening ofXRD line profile can be expressed as

β cos θ = 0.9λ/DWH + 4ε sin θ, (1)

where β is the FWHM of the Bragg peaks (in radians), θ is theBragg angle of the analysed peak and λ is the wavelength ofthe x-ray (λ = 0.154 056 nm for Cu Kα), DWH is the averagecrystallite size and ε is the strain. All Ge reflections within a2θ range 20◦–60◦ were used to construct a linear plot of βcosθversus 4sinθ , as shown in figure 3. The experimental datawere corrected for instrumental broadening. It is evident fromthe scattered data in figures 3(a) and (b) that the data do notentirely obey the WH formulation for different samples. Datafor nanocrystalline material often fail to obey the WH plot.The deviations are larger when the material being studied iselastically anisotropic because the residual strains affect someBragg reflections more than the others. The deviation observedin our data is due to the existence of anisotropic variation inthe residual strain. If we ignore the strain anisotropy effectto calculate the crystallite size and strain using equation (1),we obtain unreasonably large crystallite sizes as compared tothe sizes obtained from AFM and TEM images, and this isshown in figure 3(c). Further, the error bars are very largedue to large scatter of data in the WH plot. It is evident thatthe measured sizes are at least four times smaller than the sizederived from equation (1), which assumes an isotropic strainin the nanocrystallites. Similarly, strain (ε) calculated fromequation (1) is very high as compared with that estimated fromthe small shift of the Bragg peaks (2θ) after milling.

Ungar et al [13] proposed that the introduction of adislocation contrast factor in equation (1) might result in abetter fit to the data. According to Ungar et al, equation (1)can be expressed as

�K = 0.9/DU + �KD, �KD = 2eKC1/2, (2)

where �K = (2 cos θB�θB) /λ, �θB is the FWHM (inradians) of the Bragg reflections, λ is the wavelength of x-rays,DU is the average crystallite size, K = 2 sin θB/λ, e is thestrain, C is the dislocation contrast factor and �KD is thestrain contribution to line broadening, respectively. In theconventional WH plot it is assumed that �KD is either a linearor a quadratic function of K . If the dislocations are the main

2

J. Phys. D: Appl. Phys. 42 (2009) 245402 P K Giri

Figure 1. AFM and TEM images of freestanding Ge NCs: (a) AFM of Ge-5, inset shows the histogram of size distribution of Ge NCs,average size = 38.8 nm; (b) TEM of Ge-5, (c),(d) AFM of Ge-20, (e) TEM of Ge-20, inset shows HRTEM lattice image of single Ge NC,(f ) TEM of Ge-30 shows small nanocrystallites (marked regions) separated by pockets of amorphous regions.

contributions to the strain, then equation (2) with a 2nd ordercorrection can be expressed as [13]

�K = 0.9/DU + (πb2ρ/2B)1/2(KC1/2

)

+ (πb2Q/2B ′)1/2(K2C

), (3)

where b is modulus of the Burgers vector of dislocation, ρ

is the average dislocation density, C is the average contrastfactor, B and B ′ are constants. The constant B is taken as 10for a wide range of dislocation distribution [14]. Equation (3)shows that the proper scaling factor of the FWHM (�K) ofthe line profile is KC1/2 instead of merely K . The C factorincorporates the anisotropy factor in strain. This is usuallyreferred to as the modified WH method. Equation (3) shows

that if the contribution of the 2nd order term is much lesscompared with the 1st order term in (KC1/2), the plot of �K

versus KC1/2 would follow a straight line.

It has been shown that for a cubic crystal, the averagecontrast factors (C) are linear functions of the fourth orderinvariant of the hkl indices of the different reflections [14]:

C = Ch 0 0(1 − qH 2

), (4)

where H 2 = (h2k2 + h2l2 + k2l2

)/(h2 + k2 + l2

)2, Ch 0 0 =

α[1 − exp (−Ai/β)

]+ γAi + δ and Ai = 2C44/ (C11 − C12).

3

J. Phys. D: Appl. Phys. 42 (2009) 245402 P K Giri

Figure 2. The XRD pattern for milled Ge NCs showing gradualbroadening of Bragg peaks with increasing milling time, due to sizereduction and development of strain. Instrumental broadening issubtracted from measured line width for line profile analysis.

The unknown parameter q is basically related to the dislocationcontrast factor C and the q value depends on the typeof dislocations and elastic constants of the crystal. Thedislocation model of strain (ε2) is based on the contrastof dislocation varying with the relative orientation of theBurger vector of dislocations and the diffraction vector �b, �l, �g,respectively [14]. The dislocation contrast factor dependson the possible combination �b, �l, �g and on the anisotropicelastic constants. For cubic material, the contrast factor Ch 0 0depends on the Burger and line vectors characterizing thedislocations, the elastic anisotropy Ai and the ratio C12/C44,where C11, C12 and C44 are the elastic constants. For Ge,C12/C44 = 0.605 and Ai = 1.651 [18]. We use a computerprogram to calculate the values of Ch 0 0 for different types ofedge and screw dislocations using the known elastic constantsfor Ge. For edge dislocations we find that α = 0.1425, β =1.642, γ = 0.0198, δ = 0.0947, and for screw dislocations:α = 0.174, β = 1.952, γ = 0.0293, δ = 0.0662 [14].With these values, Ch 0 0 = 0.213 88 for screw dislocationand Ch 0 0 = 0.217 753 for edge dislocations. Thus, averagevalue Ch 0 0 = 0.215 82. In equation (3) combined withequation (4), the unknown parameters are q, DU and ρ. Theunknown parameter q is obtained by plotting KC1/2 versus�K and adjusting the q value such that the data points followa linear fit. In this way, we obtain q ≈ 1.79 for differentreflections. The q value of ∼1.8 or higher indicates that thedislocations are primarily of screw type in these Ge NCs [14].

Incorporating the anisotropy factor in strain, figures 4(a)–(c) show the plot of KC1/2 versus �K for samples with threedifferent NC sizes (obtained for different milling times) andit demonstrates the quality of the fit obtained for the presentdata using the above method. For comparison, both the linearand the quadratic fits are shown for the experimental data.Note that there is a change-over of curvature of the line in thequadratic fit in Ge-20, when compared with that of the Ge-5 and Ge-10. Since the q value was obtained with a linear

Figure 3. WH plots for (a) Ge-10, (b) Ge-20. The experimental datado not follow a linear fit (dashed line), as evident from scattered datapoints (symbols) in the plot. (c) Average crystallite size (DWH) andlattice strain (ε) as a function of milling time extracted from WHplots. The dashed lines are a guide to the eye. Due to inappropriatefit, the calculated size and strains with error bars are very largecompared with average size obtained from AFM and TEM analysis.

fit, with the 1st order term we extract the parameters (DU, ρ)

from the linear fit. The extracted parameters are shown infigure 4(d), and for comparison the size obtained from AFManalysis is also shown (open circles). The obtained size ofthe NCs is in close agreement with the size obtained fromAFM and TEM analyses. With the quadratic fit, the extractedsize (DU) comes out to be slightly lower for initial points(low milling times) than the values obtained with the linearfit. However, no significant changes in dislocation density(ρ) are obtained for the quadratic fit with respect to the linearfit. Note that the NCs’ size monotonically goes down withmilling time and we obtain a size of∼7.4 nm for 40 h of milling,with the chosen ball size. On the other hand, the dislocationdensity (ρ) and the corresponding strain first increase, reachs amaximum for 20 h milling with ρ = 13.2×1016 m−2 and thengradually reduce to ρ = 2.3 × 1016 m−2 for further millingup to 40 h. An increase in strain with the size reduction (dueto ball milling) is quite expected. However, the reduction instrain or dislocation density after 20 h of milling is likely tobe caused by in situ heating during prolonged milling. Asno external coolant arrangement was made during the millingprocess, local heating of the sample is quite likely that may

4

J. Phys. D: Appl. Phys. 42 (2009) 245402 P K Giri

Figure 4. The modified WH plots using equation (4) incorporatingcontribution of strain anisotropy in the XRD line profile: (a) Ge-5,(b) Ge-10 and (c) Ge-20 NCs. The experimental data (symbols) arefitted with linear (dashed line) and quadratic (solid line) function ofKC1/2. (d) Average crystallite size (DU ) and dislocation density (ρ)extracted from the linear fits are shown. For comparison, sizesobtained from AFM analysis are shown with open circles.

allow relaxation of the local structure in the NCs. However,no attempt was made to estimate the extent of such heating. Insitu heating during prolonged milling may tend to increase thegrain size due to grain growth, while the milling process willtend to reduce the grain size. Due to compensating effects,grain size reduction is slowed down at higher milling time, asshown in figure 4(d).

3.3. Direct evidence of strain from HRTEM

Direct evidence of lattice strain in individual NCs caused bydislocations is shown in figure 5(a) and (b) for Ge-5 andin figures 5(c) and (d) for Ge-30. The encircled regions infigures 5(a) and (c) clearly show high density of dislocationsin the strained NCs. Inverse FFT image of a selected regionof figure 5(a) is shown in figure 5(b) where the dislocationsare distinctly visible, as pointed out by an arrow. It is foundfrom HRTEM study that the spherical NCs are relatively lessstrained than the nonspherical NCs. Figure 5(d) shows smallGe nanocrystallites (marked regions) separated by pockets ofdisordered/amorphous regions in the Ge-30 sample. It may

Figure 5. (a) HRTEM images of the individual Ge NCs showingstrained lattice (regions marked with oval rings) for Ge-5. (b)Inverse FFT image of the strained lattice showing clear evidence ofdislocations (marked with an arrow) for Ge-5. (c) HRTEM image ofstrained lattice in Ge-30, (d) Small Ge nanocrystallites (markedwith ovals) separated by pockets of amorphous regions in Ge-30.

be noted that the optical Raman spectra of this sample showline shape broadening and up shift of the Ge Raman mode(at 300 cm−1) indicating strain in the lattice. However, dueto competing effects of phonon confinement and lattice strainon the Raman shift, an accurate estimate of strain was notattempted from the Raman data. Our results demonstrate thatthe Ungar et al method is quite powerful for XRD line profileanalysis of mechanically processed nanocrystalline powders,where one can isolate the contribution of nanocrystallite sizeand strain very accurately. This method is equally applicablefor analysing strain in embedded NCs of any species and shouldbe utilized for accurate determination of NC size and strainfrom the measured line width of the XRD pattern. In theGe NCs produced by milling, the dislocations are primarilyof screw type as determined by the q value obtained fromfitting, and HRTEM images corroborate this fact. As a resultof milling, screw type dislocations are expected to occur in aplanetary ball mill where both the balls and grinding jar rotateat a high speed. This action rapidly accelerates the grindingballs through both the centrifugal and Coriolis force. The rapidacceleration of the particles from one side of the jar to theother produces powerful impact forces between the balls andthe sample material while also providing additional grindingaction through frictional forces. Due to rotational motion, atwist is produced in the crystal and thus screw type dislocationsare formed.

3.4. DSC studies on strain relaxation

The strain (proportional to dislocation density) calculationshows that NCs produced after 20 h of milling have maximum

5

J. Phys. D: Appl. Phys. 42 (2009) 245402 P K Giri

Figure 6. DSC data of Ge-20 showing low temperature heat releaseat ∼313 ◦C and ∼398 ◦C due to strain relaxation. Upper inset showsthe differentiated DSC curve and lower inset shows HRTEM imageof an isolated NC in Ge-20.

strain. We believe that the change in the curvature of thequadratic fit of figure 4(c) is a reflection of this fact. Itmay imply that due to large strain in this sample, a kind ofstructural transformation occurs after 20 h of milling. Formilling beyond 20 h, there is a structural relaxation in GeNCs and the calculated dislocation density reduces as shown infigure 4(c). Further evidence for strain relaxation is obtainedfrom differential scanning calorimetry (DSC) measurementsas shown in figure 6. The upper inset of figure 6 shows thedifferentiated heat flow curve clearly showing low temperatureheat releases at ∼313 ◦C and ∼398 ◦C upon heating fromroom temperature up to 550 ◦C. Although the crystallizationtemperature of amorphous Ge is above the range of heatingtemperature, nanocrystalline Ge is expected to show a lowertemperature of crystallization. It has been reported that due tothe availability of large numbers of surface atoms the meltingpoint of semiconductor NCs (freestanding) is relatively lowas compared with their bulk crystal counterpart [19]. Similarconclusions have been made from the theoretical calculation[20], in agreement with the experimental observation from SiNCs [21]. In the present case, we relate the low temperatureheat release to structural relaxation in strained Ge NCs. Inthe strained NCs, the distortion in bond angle and bondlength is likely to recover during heating and the exothermicprocess is a signature of relaxation of the local structure ofGe NCs. Similar low temperature structural relaxation hasbeen reported from pure amorphous Ge produced by ionirradiation [22]. A measurable amount of the low temperatureheat release also suggests that strain is very high in the 20 hmilled Ge NCs, in agreement with the XRD line profileanalysis. There has been a report of complete amorphizationof Ge NCs after prolonged milling [23]. Our results donot suggest complete amorphization for milling up to 40 h.There is a likelihood of growing pockets of amorphousregions in the Ge NCs for long duration milling. OurHRTEM studies on Ge-30 indicate some pockets of disorderedor amorphous regions in the crystalline NCs, as shown infigure 5(d).

3.5. Discussions

It may be noted that the strain analysis based on the simpleXRD line profile presented here is applicable for bothfreestanding as well as embedded NCs, and strain anisotropymust be taken into account for precise determination of sizeand strain of small NCs. Although NCs grown by ball millingpossess large strain due to grinding action, embedded Ge NCsalso possess significant strain as reported by several groups.Hence, such XRD analysis should be applied for the embeddedNCs of any structure for proper analysis of strain and resultscan be compared with the other techniques, such as Ramanspectroscopy. While the line profile analysis is relativelystraightforward for cubic and diamond structure lattice, it ismore involved for the zinc-blend and wurzite structures. Inany case, the lattice strain has several implications for thestructural, thermal, electronic, magnetic and optical propertiesof the NCs. In particular, light emitting properties of GeNCs may be significantly affected by the lattice strain that isfound in the NCs grown by different methods. Our studies onembedded Ge NCs [24] have shown that visible light emissionfrom Ge NCs are not related to quantum confined carriersat the NCs, but rather originates from interface defects anddefects in the embedding matrix. Our preliminary studies onthe photoluminescence emission from these strained NCs haveshown violet-blue emission that could perhaps be related to thethin GeO2 shell formed on the Ge NCs. We believe that growthof strain-free isolated NCs may enable one to achieve thepredicted superior optical properties of Ge NCs as comparedwith that of Si NCs.

4. Conclusions

In conclusion, we presented a detailed analysis of strainand thermal relaxation of strain in freestanding Ge NCs ofsize ∼7 nm. Our studies reveal that individual Ge NCs areanisotropically strained due to the presence of a significantdensity of primarily screw dislocations. The dislocationdensity and corresponding strain vary with crystallite size witha maximum of both quantities for nanocrystallites producedafter 20 h of milling. Direct evidence for strain caused bydislocations in individual NC is presented from HRTEMimaging. DSC data analysis indicated that strain relaxationoccurs at elevated temperatures for these Ge NCs.

Acknowledgments

The author acknowledges the financial support from the Boardof Research in Nuclear Sciences (BRNS), Department ofAtomic Energy (DAE), to carry out part of this work. Hethanks D K Singh and S Kumari for help in performing someof the experiments.

References

[1] Kanjilal A, Rebohle L, Baddela N K, Zhou S, Voelskow M,Skorupa W and Helm M 2009 Phys. Rev. B 79 161302

Kolobov A V, Wei S Q, Yan S, Oyanagi H, Maeda Y andTanaka K 2003 Phys. Rev. B 67 195314

6

J. Phys. D: Appl. Phys. 42 (2009) 245402 P K Giri

Choi W K, Ng V, Ng S P, Thio H H, Shen Z X and Li W S1999 J. Appl. Phys. 86 1398

Skorupa W, Rebohle L and Gebel T 2003 Appl. Phys. A76 1049

Weissker H-Ch, Furthmuller J and Bechstedt F 2002 Phys.Rev. B 65 155327

Takeoka S, Fujii M, Hayashi S and Yamamoto K 1998 Phys.Rev. B 58 7921

[2] Araujo L L, Giulian L, Sprouster D J, Schnohr C S,Llewellyn D J, Kluth P, Cookson D J, Foran G J andRidgway M C 2008 Phys. Rev. B 78 094112

[3] Zheng F, Choi W K, Lin F, Tripathy S and Zhang J X 2008J. Phys. Chem. C 112 9223

[4] Sharp I D and Yi D O 2005 Appl. Phys. Lett. 86 063107[5] Wellner A, Paillard V, Bonafos C, Coffin H, Claverie A,

Schmidt B and Heinig K H 2003 J. Appl. Phys.94 5639

[6] Choi W K, Chew H G, Zheng F and Chim W K 2006 Appl.Phys. Lett. 89 113126

Chew H G, Zheng F, Choi W K, Chim W K, Foo Y L andFitzgerald E A 2007 Nanotechnology 18 065302

[7] Liu L, Teo k L, Shen Z X, Sun J S, Ong E H, Kolobov A V andMaeda Y 2004 Phys. Rev. B 69 125333

[8] Bernardi A, Alonso M I, Reparaz J S, Goni A R,Lacharmoise P D, Osso J O and Garriga M 2007Nanotechnology 18 475401

[9] Cheung A, Azevedo G de M, Glover C J, Llewellyn D J,Elliman R G, Foran G J and Ridgway M C 2004 Appl. Phys.Lett. 84 278

Sharp I D, Xu Q, Yi D O and Yuan C W 2006 J. Appl. Phys.100 114317

Pizzagalli L, Galli G, Klepeis J E and Gygi F 2001 Phys. Rev.B 63 165324

[10] Sharp I D, Xu Q, Liao C Y and Yi D O 2005 J. Appl. Phys.97 124316

[11] Pawlak B J, Gregorkiewicz T, Ammerlaan C A J,Takkenberg W, Tichelaar F D and Alkemade P F A 2001Phys. Rev. B 64 115308

[12] Glushenkov A M, Zhang H Z, Zou J, Lu G Q and Chen Y2007 Nanotechnology 18 175604

[13] Ungar T and Borbely A 1996 Appl. Phys. Lett. 69 3173Ungar T and Tichy G 1999 Phys. Status Solidi a 171 425

[14] Ungar T, Dragomir I, Revesz I and Borbely A 1999 J. Appl.Crystallogr. 32 992

[15] Shen T D, Schwarz R B and Thompson J D 2005 Phys. Rev. B72 014431

[16] Balogh L, Ribarik G and Ungar T 2006 J. Appl. Phys.100 023512

[17] Williamson G K and W H Hall 1951 Acta Metall. 1 22[18] Martienssen W and Warlimont H (ed) 2005 Springer

Handbook of Condensed Matter and Materials Data(Berlin: Springer) p 580

[19] Goldstein A N, Echerand C M and Alivisatos A P 1992Science 256 1425

[20] Shrivastava K N 2002 Nano Lett. 2 519[21] Goldstein A N 1996 Appl. Phys. A 62 331[22] Donovan E P, Spaepen F, Turnbull D, Poate J M and

Jacobson J D 1985 J. Appl. Phys. 57 1795[23] Gaffet E 1991 Mater. Sci. Engg. A 136 161[24] Giri P K, Bhattacharyya S, Kumari S, Das K, Ray S K,

Panigrahi B K and Nair K G M 2008 J. Appl. Phys.103 103534

7