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Quantitative description of microstructure defects in hexagonal boron nitrides using X-ray diffraction analysis
M A TE RI A L S CH A RACT ER IZ A TI O N 86 (2 0 1 3 ) 1 9 0–1 9 9
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Quantitative description of microstructure defects in hexagonal boron nitrides using X-ray diffraction analysis C. Schimpf ⁎, M. Motylenko, D. Rafaja Institute of Materials Science, TU Bergakademie Freiberg, G.-Zeuner-Str. 5, 09599 Freiberg, Germany
AR TIC LE D ATA
ABSTR ACT
Article history:
A routine for simultaneous quantification of turbostratic disorder, amount of puckering and
Received 11 April 2013
the dislocation and stacking fault density in hexagonal materials was proposed and tested on
Received in revised form
boron nitride powder samples that were synthesised using different methods. The routine
4 September 2013
allows the individual microstructure defects to be recognised according to their effect on the
Accepted 10 September 2013
anisotropy of the X-ray diffraction line broadening. For quantification of the microstructure defects, the total line broadening is regarded as a linear combination of the contributions from the particular defects. The total line broadening is obtained from the line profile fitting. As
Keywords:
testing material, graphitic boron nitride (h-BN) was employed in the form of hot-isostatically
Microstructure Defects
pressed h-BN, pyrolytic h-BN or a h-BN, which was chemically vapour deposited at a low
X-ray diffraction
temperature. The kind of the dominant microstructure defects determined from the
Line broadening
broadening of the X-ray diffraction lines was verified by high resolution transmission electron
Transmission electron microscopy
microscopy. Their amount was attempted to be verified by alternative methods.
Hexagonal BN
1.
Introduction
Boron nitride (BN) is used in a variety of technical applications. This is true for all major phases of boron nitride, i.e., for graphitic BN crystallizing within the space group P63/mmc (h-BN), for wurtzitic BN crystallizing within the space group P63mc (w-BN) and for cubic BN crystallizing within the space group F43m (c-BN). The most important properties of h-BN are its excellent temperature and thermo-shock stability, bad wettability by most metals and high electrical resistance [1]. Cubic BN and c-BN/w-BN nanocomposites are esteemed for their very high hardness, see [2,3], and references therein. Frequently, the superhard phases (c-BN and w-BN) are produced via combined high pressure and high temperature
⁎ Corresponding author. Tel.: + 49 3731 39 2094; fax: +49 3731 39 2604. E-mail address: [email protected] (C. Schimpf). 1044-5803/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.matchar.2013.09.011
© 2013 Elsevier Inc. All rights reserved.
(HP/HT) synthesis, for which h-BN is used as the starting material [4–8]. However, many authors reported that the microstructure of the h-BN precursor and in particular the kind and the density of microstructure defects in the starting h-BN strongly influences the kinetics of the phase transition and consequently the properties of the synthesis product [7,9–11]. The effect of the initial microstructure of h-BN on the properties of the BN nanocomposites is especially critical if BN nanocomposites with defined phase composition should be synthesised [2,3]. The prominent microstructure defects in h-BN are (i) turbostratic disorder, (ii) puckering, (iii) dislocations and (iv) stacking faults [12,13]. The detailed analysis of the microstructure in h-BN is usually performed with the aid of the transmission electron microscopy (TEM), which reveals the characteristics of
M A TE RI A L S C HA RACT ER I ZA TI O N 86 ( 20 1 3 ) 1 9 0–1 9 9
single defects with a very high reliability. However, as several microstructure defects can occur concurrently, a frequent task for the microstructure analysis is to quantify the density of individual microstructure defects. When defect densities averaged over large sample volume are of interest, the X-ray diffraction (XRD) is the preferred experimental method. XRD recognises the microstructure defects according to the broadening and shape of the X-ray diffraction lines (see, e.g., [14]). Especially, the amount of XRD line broadening and the dependence of the XRD line broadening on the crystallographic direction (i.e. the anisotropy of the XRD line broadening) are the most important fingerprints of the individual microstructure defects. For the prominent microstructure defects in h-BN summarised above, the anisotropy of the XRD line broadening is discussed in Section 2. Based on the different anisotropy of the XRD line broadening that is caused by turbostratic disorder, puckering, dislocations and stacking faults in h-BN, a two-stage routine for simultaneous quantification of the respective defect densities was developed and is presented in this contribution. In the first step, the XRD line broadening is obtained from the fitting of individual XRD lines by analytical functions. In the second step, the dependence of the XRD line broadening on diffraction indices, which comprise both the magnitude of the diffraction vector and the crystallographic direction, is used to identify and to quantify the relevant microstructure defects. The capability of this approach is illustrated on hot-pressed h-BN, pyrolytic h-BN and chemically vapour deposited h-BN. It is shown that this approach allows the individual microstructure defects to be identified with a high reliability and their contributions to the total line broadening to be quantified.
2. Microstructure Defects in Hexagonal BN and Their Effect on XRD Patterns The TEM investigations performed by Huang & Zhu [13] and Turan & Knowles [15] on different h-BN samples indicated that turbostratic disorder, puckering, dislocations and stacking faults are the most important microstructure defects in the graphitic boron nitride. Although only few papers dealing with the investigation of the microstructure defects in h-BN using XRD were published, the theoretical description of the XRD line broadening in hexagonal materials is available for the most relevant microstructure defects. In the following subsections, the most important results from literature are summarised. As no quantitative description of the XRD line broadening caused by puckering was available, a formula was suggested that relates the line broadening caused by the puckering to the mean elastic lattice distortion in the vicinity of the corrugated lattice planes. For the sake of uniformity, the integral line broadening (βhkl) is expressed in the reciprocal space units (dhkl⁎) throughout the text, i.e., −1 βhkl ¼ Δdhkl ¼ Δdhkl
ð1Þ
and plotted as a function of the magnitude of the diffraction vector, 4π ! sinθ ¼ 2πdhkl : q ¼ q ¼ λ
ð2Þ
191
In Eq. (1), dhkl is the interplanar spacing of the lattice planes with the diffraction indices hkl and dhkl⁎ the corresponding reciprocal lattice spacing. In Eq. (2), λ is the wavelength of the X-rays and θ the half of the diffraction angle (2θ). For hexagonal crystals, dhkl⁎ and dhkl are related to the lattice parameters a and c and to the diffraction indices hkl like: dhkl
−1 ≡ dhkl
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 4 2 2 ¼ h þ hk þ k þ 2 : 2 3a c
ð3Þ
Using the Bragg equation, qdhkl ¼ 2π
ð4Þ
and Eq. (2), the line broadening from Eq. (1) can alternatively be written in the form: βhkl ¼
Δq cosθ ¼2 Δθ 2π λ
ð5Þ
where Δθ is the line width in radians.
2.1.
Turbostratic Disorder
According to the original definition by Biscoe & Warren [16], the term turbostratic disorder is used to describe a specific defect structure in layered materials (e.g., graphite, h-BN or clay minerals), whose basal planes are mutually rotated around their normal direction and stacked in an out-of-plane direction. Particularly in h-BN, the turbostratic disorder means a regular stacking of randomly twisted {001} planes in the [001] direction. The turbostratic defects broaden predominantly the hkl lines with (h ≠ 0 or k ≠ 0) and l ≠ 0 [12,17]. The diffraction lines 00l are not affected by the turbostratic disorder, i.e., βturb 00l = 0, because the rotation of the lattice planes {001} around their normal direction does not change the interplanar spacing d00l. According to Kurdyumov [12], the broadening of the XRD lines, except 00l, that is caused by the turbostratic disorder of the hexagonal lattice planes {00l} can be described as: βturb hkl ¼
2l γ pffiffiffiffiffiffiffiffiffi : πc2 dhkl 1−γ
ð6Þ
In Eq. (6), γ ∈ 〈0,1〉 is the amount of the turbostratic disorder. The structure is perfectly ordered if γ = 0 and fully disordered if γ = 1. In Fig. 1a, the effect of the turbostratic disorder on the XRD line broadening is illustrated on h-BN with the lattice parameters a = 2.5044 Å and c = 6.6562 Å [18], and for relatively low turbostratic disorder (γ = 0.07). For a high degree of the turbostratic disorder, the diffraction lines h0l and hkl get extremely broadened. Together with a pronounced asymmetry of the h00 and hk0 lines to higher diffraction angles [12,17], the extreme broadening of the diffraction lines h0l and hkl impedes the separation of individual diffraction lines. Consequently, the series of diffraction lines with fixed h and k and with variable l can only be recognised as h0 and hk bands [12,16,17].
2.2.
Puckering
Kurdyumov et al. [19] originally defined puckering in h-BN as a non-correlated waviness of individual basal planes, which accompanies the transition of the sp2 bonds in h-BN into the
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the line broadening [Eq. (1)] due to puckering can be calculated from the differential form of Eq. (3): βpuck hkl
2 3 2 2 2 4jΔaj h þ hk þ k Δc 1 j jl ¼4 þ 3 5 : c 3a3 dhkl
ð7Þ
The above assumption about the dimension of the waviness of the basal planes was justified with the aid of TEM (see Section 5). It was discussed above that the variation of the interplanar spacing d00l in h-BN is expected to be much larger than the variation of the interplanar spacing dhk0. As d00l depends only on the lattice parameter c and dhk0 on the lattice parameter a [cf. Eq. (3)], the variation of the lattice parameter c is much larger than the variation of the lattice parameter a. Thus, Δc ≫ Δa can be anticipated in Eq. (7). In hexagonal materials, the in-plane and the out-of-plane lattice deformations are interrelated by the elastic constants C33 and S13 [20]: Δa ¼ C 33 S13 Δc:
ð8Þ
As S13 is approaching zero for h-BN [21] (see Table 1), Δa can be neglected. Thus in h-BN, the hkl-dependent XRD line broadening caused by puckering can be written in the following compact form: ¼ βpuck hkl
Fig. 1 – Dependence of the integral line broadening on the magnitude of the diffraction vector and on the diffraction indices as calculated for turbostratic disorder γ = 0.07 (a), puckering |Δc| = 0.05 Å (b), edge dislocations {001}〈110〉 with M2ρ = 5 × 1013 m−2 (c) and stacking faults with the density αsf = 0.015 (d). Open symbols represent the calculated line broadening, while the solid and the dashed lines are a guide to the eye highlighting the dependence of the line broadening on l and (h2 + hk + k2), respectively.
sp3 bonds in w-BN and c-BN. Thus, the puckering is considered as an important mechanism facilitating the transformation of h-BN to w-BN and c-BN [13,15]. Moreover, it also preserves the orientation relationships [110]h-BN||[110]w-BN||[− 110]c-BN and (002)h-BN||(002)w-BN||(111)c-BN [15]. In view of unequal differences in the interplanar spacings in the out-of-plane and in-plane directions, d002(h-BN) = 3.331 Å, d002(w-BN) = 2.114 Å, d111(c-BN) = 2.087 Å, d110 (h-BN) = 1.252 Å, d110(w-BN) = 1.277 Å and d110(c-BN) = 1.278 Å, a much larger variation of the interplanar spacing due to the puckering can be assumed for d00l(h-BN) than for dhk0(h-BN). The line broadening caused by the puckering in h-BN can be derived from the change of the reciprocal interplanar spacing [Eq. (1)]. Assuming that the waviness of the basal planes caused by the puckering extends laterally over several elementary cells, the variation of the interplanar spacing can be related to the variation of the lattice parameters. Hence,
2
jΔcjl : c3 dhkl
ð9Þ
The dependence of the XRD line broadening on the diffraction indices from Eq. (9) is shown in Fig. 1b for a = 2.5044 Å and c = 6.6562 Å [18], and for |Δc| = 0.05 Å. As compared to the XRD line broadening from turbostratic defects [Eq. (6) and puck Fig. 1a], βhkl depends much more strongly on l [cf. Eq. (9)]. Evidently, puckering broadens mainly the XRD lines 00l.
2.3.
Dislocations
The theory of the dislocation line broadening was elaborated by Krivoglaz [22] and Wilkens [23], who showed that dislocations generate microstrain, which leads to anisotropic XRD line broadening. The magnitude of the line broadening depends on the Burgers vector b, on the Wilkens factor M, on the contrast factor C hkl and on the dislocation density ρ. According to Ungár et al. [24,25], the XRD line broadening caused by dislocations can be expressed as: βdisl hkl ¼
rffiffiffi qffiffiffiffiffiffiffiffi π pffiffiffi M ρb C hkl dhkl : 2
ð10Þ
The factor M depends on the dislocation arrangement [22,23]. The contrast factor of dislocations (C hkl ) describes the dependence of the dislocation line broadening on the
Table 1 – Elastic constants of h-BN taken from Ref. [21]. h-BN C11 C12 C13 C33 C44
[GPa] [GPa] [GPa] [GPa] [GPa]
811 169 0 27 7.7
± ± ± ± ±
12 24 3 0.5 0.5
M A TE RI A L S C HA RACT ER I ZA TI O N 86 ( 20 1 3 ) 1 9 0–1 9 9
crystallographic direction. For hexagonal materials, the contrast factors were firstly derived by Klimanek & Kužel [26] and Kužel & Klimanek [27] directly from the elastic constants. Later on, Ungár & Tichy [24] and Dragomir & Ungár [28] have suggested and proven a compact formula that describes the contrast factors of dislocations in hexagonal materials with the aid of the hexagonal invariant:
C hkl
h i 2 2 2 2 2β h þ hk þ k þ γl l ¼αþ : 9 4 adhkl 4
ð11Þ
The combination of Eqs. (10) & (11) results in the following dependence of the dislocation line broadening on the diffraction indices
βdisl hkl
h i 91 8 2 2 2 2 2 rffiffiffi 2β h þ hk þ k þ γl l = π pffiffiffi< 2 bM ρ α dhkl þ ¼ : 2 9 : ; 2 ad 4
ð12Þ
hkl
Huang & Zhu [13] showed by means of the TEM investigations that the dominant dislocations in h-BN have the Burgers vector 〈110〉 and the slip system {001}. For the edge dislocations {001}〈110〉, the respective coefficients in Eqs. (11) & (12) are α = 0.02341, β = 1.6309 and γ = 1.7 × 10−4 as calculated from the average dislocation contrast factors summarised in Table 2. These contrast factors from Table 2 were obtained using the routine from Borbély et al. [29] (see also metal.elte.hu/anizc), for the elastic constants published by Bosak et al. [21], Table 1 and the lattice parameters a =2.5044 Å and c = 6.6562 Å [18]. The dependence of the dislocation XRD line broadening on the diffraction indices is illustrated in Fig. 1c for M2ρ = 5 × 1013 m− 2 and b = |〈110〉|a =2.504 Å. It can be seen that the diffraction lines 00l and hk0 are nearly unaffected by the edge
Table 2 – Averaged contrast factors of the edge dislocations {001}<110> as calculated for h-BN using the elastic constants from Table 1 by employing the computer programme provided by Borbély et al. [29]. hkl
C hkl
002 100 101 102 004 103 104 110 112 105 006 200 201 202 114 106 203 204 107 205 008 116 210
0.0277 0.02293 0.6905 1.63323 0.0277 1.9468 1.81973 0.0229 0.85873 1.55513 0.0277 0.02293 0.21667 0.6905 1.79963 1.29093 1.22053 1.63323 1.0671 1.86867 0.0277 1.92027 0.0229
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dislocations {001}〈110〉. Other diffraction lines get broader mainly with increasing l; the increase of the line broadening with increasing l is more rapid at larger values of (h2 + hk + k2).
2.4.
Stacking Faults
Another source of the anisotropic XRD line broadening in h-BN is a random stacking faulting of the {00l} planes. According to Warren [30], the randomly distributed stacking faults cause a fragmentation of coherently scattering domains (crystallites) in the 〈001〉 direction, thus for X-ray diffraction the crystallites appear smaller in the 〈001〉 direction than in the 〈hk0〉 directions. The XRD line broadening from these sf stacking faults, βhkl, increases linearly with increasing stacking fault probability αsf: βsfhkl ¼ 3
lα sf : c2 dhkl
ð13Þ
In contrast to the turbostratic defects, cf. Eq. (6) and Fig. 1a, only the widths of the diffraction lines with h − k = 3n ± 1 are affected by the randomly distributed stacking faults. The diffraction lines with h − k = 3n are not affected by these stacking faults. However, they can still be broadened because of a small crystallite size. The line broadening as calculated for a = 2.5044 Å and c = 6.6562 Å [18], and for αsf = 0.015 is displayed in Fig. 1d.
3. Identification and Quantification of the Microstructure Defects The different dependencies of the XRD line broadening on the diffraction indices hkl are summarised in Eqs. (6), (9), (12), & (13) and shown in Fig. 1. They indicate that the microstructure defects in hexagonal BN can be distinguished from each other when a sufficient number of diffraction lines is measured and analysed. The turbostratic disorder broadens mainly the diffraction lines hkl with high l; the diffraction lines hk0 and 00l are not affected. The puckering broadens mainly the diffraction lines 00l; the diffraction lines hk0 are unaffected. The edge dislocations {001}〈110〉 broaden in principle all diffraction lines hkl with higher l. However, the broadening of the diffraction lines 00l is negligible, as the Burgers vectors of these dislocations are perpendicular to the normal direction of the {001} lattice planes. The stacking faults broaden only the diffraction lines with h − k = 3n ± 1; other diffraction lines are unaffected. Turbostratic disorder and stacking faults are distinguishable on the basis of their different line broadening effects on the 11l lines. In Fig. 1, the line broadening caused by the individual microstructure defects is highlighted by solid lines for constant diffraction indices h and k, and by dashed lines for constant l. The trends observed for individual microstructure defects are unique. This allows the amount of turbostratic disorder, the magnitude of puckering, the dislocation density and the density of stacking faults to be quantified simultaneously. As the XRD line broadening depends linearly on the parameters that quantify the amount of the respective microstructure defects, i.e., on γ (1 − γ)−½, |Δc|, Mρ½ and αsf, see Eqs. (6), (9), (12) and (13), the total integral XRD line broadening can be approximated by a linear combination of the individual contributions. In this linear combination, the above parameters
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have the meaning of the weighting coefficients. Consequently, these parameters can be obtained from the measured (physical) line broadening by solving the equation MTij Mij P j ¼ MTij Bi
ð14Þ
which is equivalent to the least-squares refinement of the unknown microstructure parameters (Pj) from the physical line broadening (Bi). The matrix Mij is composed of column vectors that represent the contributions of the individual microstructure defects to the XRD line broadening. The rows of the matrix Mij stand for individual hkl reflections. The elements of Mij with respect to the different defect types (j = 1…4) are 2l Mi1 ¼ 2 πc dhkl
The upper index T in Eq. (14) means the transposed matrix. Pj is a column vector containing the refined parameters described above in the form: Pj ¼
γ pffiffiffiffiffiffiffiffiffi jΔcj 1−γ
pffiffiffi M ρ α sf
1 D
T :
ð20Þ
D in Eq. (20) is the crystallite size. The column vector Bi in Eq. (14) contains the measured integral breadths of the diffraction lines hkl that were corrected for the instrumental line broadening. The quality of the least-square fit [Eq. (15)] was judged by using the following goodness of fit:
ð15Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 uX exp calc u u i βi −βi t Rw ¼ 100% X exp 2 βi i
ð16Þ
= MijPj. The estimated standard deviations of the with βcalc i refined microstructure parameters were calculated from the diagonal elements of the inverse matrix (MTijMij)−1,
ð22Þ
for turbostratic disorder, 2
l Mi2 ¼ 3 c dhkl for puckering defects,
h i 91 8 2 2 2 2 2 rffiffiffi < 4 3:2618 h þ hk þ k þ 1:7 10−4 l l = π b 0:02341 þ Mi3 ¼ 4 ; 2 : 9a2 c2 d
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X exp calc 2 u
−1 u βi −βi i : σ j ¼ t MTij Mij n−p jj
ð23Þ
hkl
ð17Þ Here, n and p are the number of experimentally determined line widths and the number of refined parameters, respectively.
for the {001}〈110〉 edge dislocations, and Mi4 ¼
3l for h–k ¼ 3n 1 c2 dhkl
ð18Þ
Mi4 ¼ 0 for h–k ¼ 3n for stacking faults on basal planes. The last column in the Mij matrix is devoted to a constant XRD line broadening due to the small crystallites (according to Scherrer [31]) and reads Mi5 ¼ 1:
ð19Þ
4.
Materials and Methods
The routine described above was tested on samples of graphitic boron nitride containing the microstructure defects discussed above in different concentrations. Hot-pressed h-BN (HDBN) was purchased from HENZE Boron Nitride Products (HeBoSint®) in rod shape. It is characterised by a low
Fig. 2 – X-ray diffraction patterns of different h-BN samples: hot-pressed high-density boron nitride (HDBN), pyrolytic boron nitride (pBN) and chemically vapour deposited boron nitride (CVD-BN). Open circles represent the measured data, and the solid lines the result of the line profile fitting. The intensities are plotted in logarithmic scale. Individual XRD patterns were mutually shifted for clarity. In regions with significant overlap of heavily broadened diffraction maxima, individual peaks are indicated by dashed lines. The diffraction indices are displayed at the top of the graph.
M A TE RI A L S C HA RACT ER I ZA TI O N 86 ( 20 1 3 ) 1 9 0–1 9 9
concentration of microstructure defects and a good crystallinity. Pyrolytic h-BN (pBN) was purchased from Morgan Technical Ceramics. This material is produced in a high temperature CVD process and has a high density of puckering defects. Another CVD h-BN (CVD-BN), which was kindly provided by Dr. I. Petrusha (Institute of Superhard Materials at the National Academy of Sciences of Ukraine in Kiev), has been chemically vapour deposited at a low deposition temperature. This material is sometimes referred to as “turbostratic BN” because it has a very high density of turbostratic defects. All investigated materials fulfil the conditions for powder diffraction. The X-ray diffraction experiments were performed in Bragg– Brentano geometry using a Seifert FPM URD6.5 diffractometer that was equipped with a sealed X-ray tube with Cu anode and with a graphite monochromator located in the diffracted beam. The instrumental line broadening was determined using LaB6 purchased from NIST (SRM660b) [32]. Interpolation of the instrumental line broadening function was done with the equation postulated by Louër & Langford [33]. The experimental integral breadths were extracted from the measured diffraction patterns by using line profile fitting. Individual diffraction lines were approximated by an asymmetric Pearson VII function [34]. The physical line broadening was obtained by subtracting the instrumental line broadening from the measured integral breadths as usual for the Cauchy (Lorentz) shape of the diffraction lines [35]. The results obtained from XRD, in particular the crystallite sizes and the kind of the dominant microstructure defects, were verified by TEM and/or high resolution TEM (HRTEM). The (HR) TEM experiments were performed using a JEM 2200FS field emission TEM (Jeol), which is equipped with a Cs corrector in the primary beam (Ceos). The transmission electron microscope was operated at 200 kV acceleration voltage. TEM preparation was done by carefully cutting slices from the bulk starting materials. These were ground and mechanically polished. The final preparation step was the Ar+ ion thinning of the samples.
5.
identified that the stacking faults on the {001} planes also contribute to the XRD line broadening. The parameters obtained from the refinement are M2ρ = (6.6 ± 0.4) × 1012 m−2 and αsf = (4 ± 1) × 10−4. Using the first approximation of the pffiffiffiffiffi−1 Wilkens factor, M≈ ln b ρ0 ≈7 [23], the true dislocation density in HDBN was estimated as ρ ≈ 1011 m−2. The amount of the puckering was very low, Δc = (0.2 ± 0.1) × 10−3 Å. The degree of the turbostratic disorder was equal to zero within the accuracy of the least-square fit [Eq. (14)] and therefore neglected. This confirms the results of other authors, who did not observe turbostratic disorder in h-BN after hot-isostatic pressing [36]. The absence of the turbostratic disorder in HDBN can also be concluded from the absence of the asymmetry of the line shape of the h00 and hk0 diffraction lines (Fig. 2).
Results
The XRD patterns of h-BN with different densities of microstructure defects are shown in Fig. 2. HDBN with the lowest defect density produces the narrowest but still anisotropically broadened diffraction lines. The high density of puckering defects in pBN is apparent from the large broadening of the diffraction lines 00l. The high density of turbostratic defects in CVD-BN causes a rapid increase of the broadening of the diffraction lines 10l and 11l with increasing l. Therefore, only the 10 and 11 bands are visible. This phenomenon is illustrated on the diffraction lines 100, 101, 102 and 103 (dashed lines in Fig. 2). In contrast, the diffraction lines 00l are well pronounced, as they are not affected by the turbostratic defects (see Section 2.1).
5.1.
195
Hot-pressed High-density Boron Nitride (HDBN)
The large broadening of the diffraction lines 10l and a moderate broadening of the diffraction lines 00l (Fig. 3a) suggest that the edge dislocations {001}〈110〉 are the most dominant microstructure defects in HDBN (cf. Fig. 1c). The analysis of the microstructure defects with the aid of Eq. (14)
Fig. 3 – Dependence of the integral line widths on the magnitude of the diffraction vector as obtained from the line profile fitting (circles) and as calculated from the microstructure parameters determined using Eq. (14) (large crosses). Small crosses mark the line widths calculated for overlapping, very weak or extremely broadened diffraction lines, whose experimental counterparts were inaccessible. The line oscillating around 0 shows the differences between experimental and calculated data. The weighted reliability factor (Rw) was determined using Eq. (22).
196
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The constant part of the line broadening was quite low. This indicates a large crystallite size, which was calculated to be D = (260 ± 15) nm. This crystallite size was confirmed by TEM. The dark-field TEM micrograph (Fig. 4a) shows exemplarily a grain with the size between 250 and 350 nm. Also the dislocations were proven to be the most frequent microstructure defects in HDBN. In Fig. 4b, typical dislocation structures can be seen in the dark-field image that was recorded at the 224 diffraction spot in order to enhance the contrast of the dislocations with the Burgers vector 〈110〉.
5.2.
Pyrolytic Boron Nitride (pBN)
The anisotropy of the line broadening observed in pBN (Fig. 3b) is characteristic for graphitic boron nitride containing puckered basal planes (cf. Fig. 1b). Accordingly, a large magnitude of puckering, |Δc| = (0.19 ± 0.02) Å, has been determined with the aid of Eq. (14). Further microstructure defect that significantly contributes to the line broadening in pBN is the turbostratic disorder with the magnitude of γ = 0.14 ± 0.01. The crystallites in this materials are small, D = (14 ± 2) nm. There was no noteworthy contribution of dislocations and
stacking faults to the observed line broadening, thus the respective refinable parameters of these microstructure defects were regarded to be zero within the accuracy of this approach. HRTEM (Fig. 5) recognises the puckering in pBN as a waviness of the lattice planes {001} and as a variation of the interplanar spacing d00l. These phenomena are visible in the Fast Fourier Transform (FFT) of the HRTEM micrograph (inset in Fig. 5) as well. The waviness of the lattice planes {001} causes the azimuthal broadening of the diffraction spots (00l) the variation of the interplanar spacing d00l, their radial broadening. Due to the considerable turbostratic disorder, only the lattice planes {001} but no pronounced electron density modulations within these lattice planes are visible. Furthermore, HRTEM verified a small size of coherently scattering domains (crystallites) that was concluded from the XRD line broadening. It is worth noting that the crystallite size determined by XRD (D ≈ 14 nm) should be compared with the crystallite size obtained by HRTEM in the [hk0] directions. In these directions, mainly the disturbances marked by dashed lines in Fig. 5 destroy the coherence between the neighbouring domains. The size of the continuous regions ranges between 10 and 20 nm; this agrees very well with the crystallite size obtained from the XRD line broadening. In the [001] direction, the condition for coherent scattering between the subsequent {00l} lattice planes is mainly disturbed by the puckering, which is also the main source of the broadening of the XRD lines (00l), cf. Fig. 3b.
5.3.
Chemical Vapour Deposited Boron Nitride (CVD-BN)
In boron nitride that was chemically vapour deposited at low temperatures (CVD-BN), the turbostratic disorder, puckering and small crystallite size have been identified as the main contributions to the line broadening. The fitting of the integral
Fig. 4 – (a) Two partly overlapping h-BN grains in HDBN as seen using dark-field TEM. Their size agrees well with the mean crystallite size determined using XRD (approximately 260 nm). (b) Typical dislocation arrangement observed in the dark-field contrast. This TEM micrograph was taken using theh 224 i diffraction spot. The direction of the primary beam is 111 .
Fig. 5 – HRTEM micrograph of pBN showing the puckering (waviness) of the basal layers (lattice planes {001}). Regions with (apparently) continuous basal layers are separated by highly disturbed regions, where the basal layers appear discontinuous (dashed lines). The FFT of the HRTEM image is shown in the inset.
197
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6.
Discussion
6.1. Reliability of the Defect Densities Determined Using the Presented Method
Fig. 6 – HRTEM micrograph of CVD-BN and FFT/HRTEM (inset) illustrates a very small crystallite size (D ≈ 5 nm) and a locally wide orientation distribution of the 〈001〉 directions. The dashed lines mark the lattice planes {001}.
breadths from Fig. 3c using the microstructure model from Section 3 revealed the amount of turbostratic disorder γ = 0.73 ± 0.02, the puckering |Δc| = (0.12 ± 0.05) Å and the crystallite size D = (5 ± 1) nm. Similar to pBN, the contribution of stacking faults and dislocations to the overall line broadening observed in CVD-BN is too small to be reasonably quantified from the experimental data shown in Fig. 3c. The small crystallite size concluded from the XRD line broadening was confirmed by HRTEM, which revealed the grain size of approximately 5 nm (Fig. 6). Obviously, CVD-BN has a lower {001} local preferred orientation of crystallites than pBN. This can be seen mainly from different azimuthal broadenings of the diffraction spot (002), which are wider in CVD-BN than in pBN (compare FFT of the HRTEM micrographs in Figs. 5 and 6). The different local orientations of crystallites in CVD-BN and pBN are also indicated by the intensity ratios I(002)/I(100) and I(002)/I(110) that are higher in pBN than in CVD-BN (see Fig. 2). Because of the smaller amount of puckering in CVD-BN as compared with pBN, the interplanar distances d00l vary less in CVD-BN than in pBN. Both XRD and FFT/HRTEM recognise the smaller variance of d00l in CVD-BN as a smaller (radial) broadening of the diffraction lines 00l, see Figs. 3, 5 and 6. Like in pBN, the very high amount of the turbostratic disorder in CVD-BN makes the visualisation of the electron density modulations using HRTEM within the {00l} lattice planes impossible.
As already discussed above, transmission electron microscopy confirmed the presence of the dominant microstructure defects that was concluded from the XRD measurements. In order to be able to judge the reliability of the defect densities calculated from the XRD line broadening, the defect densities were additionally determined using alternative approaches based on the analysis of XRD data. Since the estimation of the amount of turbostratic disorder from the XRD line broadening has the most alternative approaches available, these were used to verify the amount of turbostratic disorder (γ) obtained from Eq. (14). The results of the comparison are collected in Table 3. The first approach was developed by Kurdyumov et al. [18], who found a quadratic dependence of the amount of turbostratic disorder in h-BN on the interplanar spacing d002. In our samples, the interplanar spacing d002 increased from HDBN over pBN to CVD-BN (see Table 3). Consequently, the degree of the turbostratic disorder calculated using the approach from Ref. [18], which is referred to as method #2 in Table 3, was γ ≈ 0, γ ≈ 0.4 and γ ≈ 1 for HDBN, pBN and CVD-BN, respectively. This trend corresponds to the increase of the turbostratic disorder found by applying our method (method #1 in Table 3), but the turbostratic disorder calculated from d002 is clearly overestimated. A possible reason is the effect of puckering on the interplanar spacing d002 that is not considered in Kurdyumov's model [18]. Another alternative approach [12,18] (method #3 in Table 3) determines the degree of the turbostratic disorder from the broadening of the XRD line 112, i.e. from β112. The relationship between β112 and the amount of turbostratic disorder is actually based on Eq. (6). In order to take the instrumental line broadening and a small crystallite size into account, Kurdyumov et al. [18] suggested correcting β112 with the aid of β110. As it can be seen from Fig. 1, the diffraction line 112 is heavily broadened by turbostratic defects and edge dislocations {001}〈110〉, but it is almost not affected by stacking faults and puckering, respectively. On the other hand, apart from the instrumental line broadening and small crystallite size, the diffraction line 110 is only broadened by dislocations. This approach revealed the amount of turbostratic disorder of γ = 0.03 for HDBN, γ = 0.15 for pBN and γ = 0.71 for CVD-BN, which are values being very close to those obtained from the refinement of the β(q) plots (Fig. 3).
Table 3 – Comparison of the amounts of the turbostratic disorder (γ) calculated using our approach (method #1), using the quadratic relationship between the amount of turbostratic disorder in h-BN and the interplanar spacing d002 [18] (method #2), and using the Kurdyumov method (#3) from Refs. [12] and [18], which is briefly described in Section 6.1. Method #1 Sample HDBN pBN CVD-BN
γ 0 0.14 ± 0.01 0.73 ± 0.02
Method #2
Method #3 −1
c = 2 · d002[Å]
γ
β110[Å ]
β112[Å− 1]
γ
6.657 ± 0.001 6.696 ± 0.003 6.883 ± 0.003
≈0 ≈0.4 ≈1.0
1.6 × 10−3 8.8 × 10−3 2.5 × 10−2
2.5 × 10−3 1.5 × 10−2 7.2 × 10−2
0.03 0.15 0.71
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If dislocations are the dominant microstructure defects, the calculation of the dislocation density from the line broadening is relatively straightforward (see Section 2.3). The procedure based on Eqs. (10) to (12) was successfully used by Klimanek & Kužel [26] and Dragomir & Ungár [28] to obtain the dislocation densities in cold-worked zirconium and titanium, respectively. The main drawback of this method is that it reveals only the dislocation density multiplied by the squared Wilkens factor, i.e., M2ρ. The Wilkens factor (M) describing the interaction of the dislocation field is required if the dislocation density is to be determined independently.
6.2.
Presented Approach Versus Single-stage Routines
As explained above, the presented routine handles the XRD data in two steps. In the first step, the measured diffracted intensities are approximated by analytical functions in order to separate the individual diffraction lines from each other and to obtain their integral breadths. In the second step, the dependence of the physical integral breadths on diffraction indices (after a correction for instrumental line broadening) is analysed using Eq. (14) in order to obtain the amounts of individual microstructure defects. Nowadays, the XRD patterns are frequently processed by using single-stage routines that allow the structure and microstructure parameters to be obtained in one step. Typical representatives of such routines employ the Rietveld approach [37,38], which refines the parameters of a structure model by fitting the calculated XRD patterns to the entire measured data. In materials containing microstructure defects, the crystallographic anisotropy of the XRD line broadening was always a factor, which deteriorates the agreement between the measured and calculated XRD patterns and which consequently influences the quality of the refined structure model. Therefore, already the earlier Rietveld-like computer programmes have handled the dependence of the XRD line broadening on the diffraction indices. Many of them employ the 4th order crystallographic invariants of diffraction indices as suggested by Popa [39] and Stephens [40]. The 4th order invariants are capable of describing the anisotropy of the crystal lattice deformation caused by the strain fields of microstructure defects. Thus, the anisotropic XRD line broadening calculated using these invariants improves the quality of the Rietveld fit in many cases. However, this method provides no information about the kind and density of the microstructure defects. Later on, physical models describing quantitatively the amount of the XRD line broadening for various microstructure defects were implemented into the Rietveld routines [17,41–45]. This allowed selected microstructure features like mean crystallite size and crystallite size distribution, dislocation density, stacking fault probability and degree of the turbostratic disorder to be quantified together with the lattice parameters and preferred orientation of crystallites. However, none of these routines is capable of modelling concurrently all microstructure defects observed in h-BN, i.e., turbostratic disorder, puckering, dislocations and stacking faults. The first reason is a missing description of the XRD line broadening caused by puckering. Secondly, strong correlations between individual microstructure parameters are expected, especially when several types of microstructure defects are present in the
samples simultaneously or if the number of refined microstructure parameters is large. In this context, it is worthy of noting that the Rietveld-like routines compare whole measured and calculated XRD patterns. Thus, these methods require some additional parameters describing the line shape, thereby they increase the number of free parameters further. From this point of view, the two-stage routine based on the presented approach is much more stable than the one-stage Rietveld-like routines, especially if several concurrent microstructure defects are present in the material under study and if their contributions to the total XRD line broadening are to be quantified at the same time. On the other hand, the Rietveld-based methods provide more detailed microstructure information once the dominant kind of the microstructure defects is known and refinement constraints can be applied.
7.
Conclusions
It was shown that the most frequent microstructure defects in hexagonal boron nitride, namely turbostratic disorder, puckering, dislocations and stacking faults can be distinguished from each other according to their effects on the amount and crystallographic anisotropy of the X-ray diffraction line broadening. A routine was suggested that allows both the dominant kind of the microstructure defects to be identified and quantified by suitable parameters. For this purpose, the measured XRD line broadening is regarded as a linear combination of the unique XRD line broadening from specific microstructure defects. For the quantification of the turbostratic disorder, dislocations and stacking faults, the existing models were employed. For the quantitative description of the (uncorrelated) puckering of the basal layers, a new model was created, which assumes that the puckering leads to a variation of the lattice parameter c. The routine was tested on hot-isostatically pressed h-BN, pyrolytic h-BN and low-temperature CVD h-BN. In the hot-isostatically pressed h-BN, the dominant microstructure defects are the edge dislocations {001}〈110〉 and the stacking faults on the {001} planes; the amount of puckering is very low and the turbostratic defects are almost absent. The microstructures of the pyrolytic h-BN and low-temperature CVD h-BN are dominated by the puckering and by the presence of turbostratic defects. The puckering is the most important microstructure defect in pyrolytic h-BN, whereas the turbostratic disorder is the typical microstructure feature in CVD h-BN.
Acknowledgements This investigation was carried out in the frame of the Freiberg High Pressure Research Centre (FHP) financed through the Dr. Erich Krüger Foundation of the TU Bergakademie Freiberg. The TEM JEOL 2200 FS was purchased from the funds of the Cluster of Excellence “Structure Design of Novel High-Performance Materials via Atomic Design and Defect Engineering (ADDE)” which is supported by the European Union (European Regional Development Fund) and the Ministry of Science and Art of Saxony (SMWK). The authors thank Dr. Zdeněk Matěj (Charles University Prague) for valuable discussion.
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Quantitative description of microstructure defects in hexagonal boron nitrides using X-ray diffraction analysis C. Schimpf ⁎, M. Motylenko, D. Rafaja Institute of Materials Science, TU Bergakademie Freiberg, G.-Zeuner-Str. 5, 09599 Freiberg, Germany
AR TIC LE D ATA
ABSTR ACT
Article history:
A routine for simultaneous quantification of turbostratic disorder, amount of puckering and
Received 11 April 2013
the dislocation and stacking fault density in hexagonal materials was proposed and tested on
Received in revised form
boron nitride powder samples that were synthesised using different methods. The routine
4 September 2013
allows the individual microstructure defects to be recognised according to their effect on the
Accepted 10 September 2013
anisotropy of the X-ray diffraction line broadening. For quantification of the microstructure defects, the total line broadening is regarded as a linear combination of the contributions from the particular defects. The total line broadening is obtained from the line profile fitting. As
Keywords:
testing material, graphitic boron nitride (h-BN) was employed in the form of hot-isostatically
Microstructure Defects
pressed h-BN, pyrolytic h-BN or a h-BN, which was chemically vapour deposited at a low
X-ray diffraction
temperature. The kind of the dominant microstructure defects determined from the
Line broadening
broadening of the X-ray diffraction lines was verified by high resolution transmission electron
Transmission electron microscopy
microscopy. Their amount was attempted to be verified by alternative methods.
Hexagonal BN
1.
Introduction
Boron nitride (BN) is used in a variety of technical applications. This is true for all major phases of boron nitride, i.e., for graphitic BN crystallizing within the space group P63/mmc (h-BN), for wurtzitic BN crystallizing within the space group P63mc (w-BN) and for cubic BN crystallizing within the space group F43m (c-BN). The most important properties of h-BN are its excellent temperature and thermo-shock stability, bad wettability by most metals and high electrical resistance [1]. Cubic BN and c-BN/w-BN nanocomposites are esteemed for their very high hardness, see [2,3], and references therein. Frequently, the superhard phases (c-BN and w-BN) are produced via combined high pressure and high temperature
⁎ Corresponding author. Tel.: + 49 3731 39 2094; fax: +49 3731 39 2604. E-mail address: [email protected] (C. Schimpf). 1044-5803/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.matchar.2013.09.011
© 2013 Elsevier Inc. All rights reserved.
(HP/HT) synthesis, for which h-BN is used as the starting material [4–8]. However, many authors reported that the microstructure of the h-BN precursor and in particular the kind and the density of microstructure defects in the starting h-BN strongly influences the kinetics of the phase transition and consequently the properties of the synthesis product [7,9–11]. The effect of the initial microstructure of h-BN on the properties of the BN nanocomposites is especially critical if BN nanocomposites with defined phase composition should be synthesised [2,3]. The prominent microstructure defects in h-BN are (i) turbostratic disorder, (ii) puckering, (iii) dislocations and (iv) stacking faults [12,13]. The detailed analysis of the microstructure in h-BN is usually performed with the aid of the transmission electron microscopy (TEM), which reveals the characteristics of
M A TE RI A L S C HA RACT ER I ZA TI O N 86 ( 20 1 3 ) 1 9 0–1 9 9
single defects with a very high reliability. However, as several microstructure defects can occur concurrently, a frequent task for the microstructure analysis is to quantify the density of individual microstructure defects. When defect densities averaged over large sample volume are of interest, the X-ray diffraction (XRD) is the preferred experimental method. XRD recognises the microstructure defects according to the broadening and shape of the X-ray diffraction lines (see, e.g., [14]). Especially, the amount of XRD line broadening and the dependence of the XRD line broadening on the crystallographic direction (i.e. the anisotropy of the XRD line broadening) are the most important fingerprints of the individual microstructure defects. For the prominent microstructure defects in h-BN summarised above, the anisotropy of the XRD line broadening is discussed in Section 2. Based on the different anisotropy of the XRD line broadening that is caused by turbostratic disorder, puckering, dislocations and stacking faults in h-BN, a two-stage routine for simultaneous quantification of the respective defect densities was developed and is presented in this contribution. In the first step, the XRD line broadening is obtained from the fitting of individual XRD lines by analytical functions. In the second step, the dependence of the XRD line broadening on diffraction indices, which comprise both the magnitude of the diffraction vector and the crystallographic direction, is used to identify and to quantify the relevant microstructure defects. The capability of this approach is illustrated on hot-pressed h-BN, pyrolytic h-BN and chemically vapour deposited h-BN. It is shown that this approach allows the individual microstructure defects to be identified with a high reliability and their contributions to the total line broadening to be quantified.
2. Microstructure Defects in Hexagonal BN and Their Effect on XRD Patterns The TEM investigations performed by Huang & Zhu [13] and Turan & Knowles [15] on different h-BN samples indicated that turbostratic disorder, puckering, dislocations and stacking faults are the most important microstructure defects in the graphitic boron nitride. Although only few papers dealing with the investigation of the microstructure defects in h-BN using XRD were published, the theoretical description of the XRD line broadening in hexagonal materials is available for the most relevant microstructure defects. In the following subsections, the most important results from literature are summarised. As no quantitative description of the XRD line broadening caused by puckering was available, a formula was suggested that relates the line broadening caused by the puckering to the mean elastic lattice distortion in the vicinity of the corrugated lattice planes. For the sake of uniformity, the integral line broadening (βhkl) is expressed in the reciprocal space units (dhkl⁎) throughout the text, i.e., −1 βhkl ¼ Δdhkl ¼ Δdhkl
ð1Þ
and plotted as a function of the magnitude of the diffraction vector, 4π ! sinθ ¼ 2πdhkl : q ¼ q ¼ λ
ð2Þ
191
In Eq. (1), dhkl is the interplanar spacing of the lattice planes with the diffraction indices hkl and dhkl⁎ the corresponding reciprocal lattice spacing. In Eq. (2), λ is the wavelength of the X-rays and θ the half of the diffraction angle (2θ). For hexagonal crystals, dhkl⁎ and dhkl are related to the lattice parameters a and c and to the diffraction indices hkl like: dhkl
−1 ≡ dhkl
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2 4 2 2 ¼ h þ hk þ k þ 2 : 2 3a c
ð3Þ
Using the Bragg equation, qdhkl ¼ 2π
ð4Þ
and Eq. (2), the line broadening from Eq. (1) can alternatively be written in the form: βhkl ¼
Δq cosθ ¼2 Δθ 2π λ
ð5Þ
where Δθ is the line width in radians.
2.1.
Turbostratic Disorder
According to the original definition by Biscoe & Warren [16], the term turbostratic disorder is used to describe a specific defect structure in layered materials (e.g., graphite, h-BN or clay minerals), whose basal planes are mutually rotated around their normal direction and stacked in an out-of-plane direction. Particularly in h-BN, the turbostratic disorder means a regular stacking of randomly twisted {001} planes in the [001] direction. The turbostratic defects broaden predominantly the hkl lines with (h ≠ 0 or k ≠ 0) and l ≠ 0 [12,17]. The diffraction lines 00l are not affected by the turbostratic disorder, i.e., βturb 00l = 0, because the rotation of the lattice planes {001} around their normal direction does not change the interplanar spacing d00l. According to Kurdyumov [12], the broadening of the XRD lines, except 00l, that is caused by the turbostratic disorder of the hexagonal lattice planes {00l} can be described as: βturb hkl ¼
2l γ pffiffiffiffiffiffiffiffiffi : πc2 dhkl 1−γ
ð6Þ
In Eq. (6), γ ∈ 〈0,1〉 is the amount of the turbostratic disorder. The structure is perfectly ordered if γ = 0 and fully disordered if γ = 1. In Fig. 1a, the effect of the turbostratic disorder on the XRD line broadening is illustrated on h-BN with the lattice parameters a = 2.5044 Å and c = 6.6562 Å [18], and for relatively low turbostratic disorder (γ = 0.07). For a high degree of the turbostratic disorder, the diffraction lines h0l and hkl get extremely broadened. Together with a pronounced asymmetry of the h00 and hk0 lines to higher diffraction angles [12,17], the extreme broadening of the diffraction lines h0l and hkl impedes the separation of individual diffraction lines. Consequently, the series of diffraction lines with fixed h and k and with variable l can only be recognised as h0 and hk bands [12,16,17].
2.2.
Puckering
Kurdyumov et al. [19] originally defined puckering in h-BN as a non-correlated waviness of individual basal planes, which accompanies the transition of the sp2 bonds in h-BN into the
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the line broadening [Eq. (1)] due to puckering can be calculated from the differential form of Eq. (3): βpuck hkl
2 3 2 2 2 4jΔaj h þ hk þ k Δc 1 j jl ¼4 þ 3 5 : c 3a3 dhkl
ð7Þ
The above assumption about the dimension of the waviness of the basal planes was justified with the aid of TEM (see Section 5). It was discussed above that the variation of the interplanar spacing d00l in h-BN is expected to be much larger than the variation of the interplanar spacing dhk0. As d00l depends only on the lattice parameter c and dhk0 on the lattice parameter a [cf. Eq. (3)], the variation of the lattice parameter c is much larger than the variation of the lattice parameter a. Thus, Δc ≫ Δa can be anticipated in Eq. (7). In hexagonal materials, the in-plane and the out-of-plane lattice deformations are interrelated by the elastic constants C33 and S13 [20]: Δa ¼ C 33 S13 Δc:
ð8Þ
As S13 is approaching zero for h-BN [21] (see Table 1), Δa can be neglected. Thus in h-BN, the hkl-dependent XRD line broadening caused by puckering can be written in the following compact form: ¼ βpuck hkl
Fig. 1 – Dependence of the integral line broadening on the magnitude of the diffraction vector and on the diffraction indices as calculated for turbostratic disorder γ = 0.07 (a), puckering |Δc| = 0.05 Å (b), edge dislocations {001}〈110〉 with M2ρ = 5 × 1013 m−2 (c) and stacking faults with the density αsf = 0.015 (d). Open symbols represent the calculated line broadening, while the solid and the dashed lines are a guide to the eye highlighting the dependence of the line broadening on l and (h2 + hk + k2), respectively.
sp3 bonds in w-BN and c-BN. Thus, the puckering is considered as an important mechanism facilitating the transformation of h-BN to w-BN and c-BN [13,15]. Moreover, it also preserves the orientation relationships [110]h-BN||[110]w-BN||[− 110]c-BN and (002)h-BN||(002)w-BN||(111)c-BN [15]. In view of unequal differences in the interplanar spacings in the out-of-plane and in-plane directions, d002(h-BN) = 3.331 Å, d002(w-BN) = 2.114 Å, d111(c-BN) = 2.087 Å, d110 (h-BN) = 1.252 Å, d110(w-BN) = 1.277 Å and d110(c-BN) = 1.278 Å, a much larger variation of the interplanar spacing due to the puckering can be assumed for d00l(h-BN) than for dhk0(h-BN). The line broadening caused by the puckering in h-BN can be derived from the change of the reciprocal interplanar spacing [Eq. (1)]. Assuming that the waviness of the basal planes caused by the puckering extends laterally over several elementary cells, the variation of the interplanar spacing can be related to the variation of the lattice parameters. Hence,
2
jΔcjl : c3 dhkl
ð9Þ
The dependence of the XRD line broadening on the diffraction indices from Eq. (9) is shown in Fig. 1b for a = 2.5044 Å and c = 6.6562 Å [18], and for |Δc| = 0.05 Å. As compared to the XRD line broadening from turbostratic defects [Eq. (6) and puck Fig. 1a], βhkl depends much more strongly on l [cf. Eq. (9)]. Evidently, puckering broadens mainly the XRD lines 00l.
2.3.
Dislocations
The theory of the dislocation line broadening was elaborated by Krivoglaz [22] and Wilkens [23], who showed that dislocations generate microstrain, which leads to anisotropic XRD line broadening. The magnitude of the line broadening depends on the Burgers vector b, on the Wilkens factor M, on the contrast factor C hkl and on the dislocation density ρ. According to Ungár et al. [24,25], the XRD line broadening caused by dislocations can be expressed as: βdisl hkl ¼
rffiffiffi qffiffiffiffiffiffiffiffi π pffiffiffi M ρb C hkl dhkl : 2
ð10Þ
The factor M depends on the dislocation arrangement [22,23]. The contrast factor of dislocations (C hkl ) describes the dependence of the dislocation line broadening on the
Table 1 – Elastic constants of h-BN taken from Ref. [21]. h-BN C11 C12 C13 C33 C44
[GPa] [GPa] [GPa] [GPa] [GPa]
811 169 0 27 7.7
± ± ± ± ±
12 24 3 0.5 0.5
M A TE RI A L S C HA RACT ER I ZA TI O N 86 ( 20 1 3 ) 1 9 0–1 9 9
crystallographic direction. For hexagonal materials, the contrast factors were firstly derived by Klimanek & Kužel [26] and Kužel & Klimanek [27] directly from the elastic constants. Later on, Ungár & Tichy [24] and Dragomir & Ungár [28] have suggested and proven a compact formula that describes the contrast factors of dislocations in hexagonal materials with the aid of the hexagonal invariant:
C hkl
h i 2 2 2 2 2β h þ hk þ k þ γl l ¼αþ : 9 4 adhkl 4
ð11Þ
The combination of Eqs. (10) & (11) results in the following dependence of the dislocation line broadening on the diffraction indices
βdisl hkl
h i 91 8 2 2 2 2 2 rffiffiffi 2β h þ hk þ k þ γl l = π pffiffiffi< 2 bM ρ α dhkl þ ¼ : 2 9 : ; 2 ad 4
ð12Þ
hkl
Huang & Zhu [13] showed by means of the TEM investigations that the dominant dislocations in h-BN have the Burgers vector 〈110〉 and the slip system {001}. For the edge dislocations {001}〈110〉, the respective coefficients in Eqs. (11) & (12) are α = 0.02341, β = 1.6309 and γ = 1.7 × 10−4 as calculated from the average dislocation contrast factors summarised in Table 2. These contrast factors from Table 2 were obtained using the routine from Borbély et al. [29] (see also metal.elte.hu/anizc), for the elastic constants published by Bosak et al. [21], Table 1 and the lattice parameters a =2.5044 Å and c = 6.6562 Å [18]. The dependence of the dislocation XRD line broadening on the diffraction indices is illustrated in Fig. 1c for M2ρ = 5 × 1013 m− 2 and b = |〈110〉|a =2.504 Å. It can be seen that the diffraction lines 00l and hk0 are nearly unaffected by the edge
Table 2 – Averaged contrast factors of the edge dislocations {001}<110> as calculated for h-BN using the elastic constants from Table 1 by employing the computer programme provided by Borbély et al. [29]. hkl
C hkl
002 100 101 102 004 103 104 110 112 105 006 200 201 202 114 106 203 204 107 205 008 116 210
0.0277 0.02293 0.6905 1.63323 0.0277 1.9468 1.81973 0.0229 0.85873 1.55513 0.0277 0.02293 0.21667 0.6905 1.79963 1.29093 1.22053 1.63323 1.0671 1.86867 0.0277 1.92027 0.0229
193
dislocations {001}〈110〉. Other diffraction lines get broader mainly with increasing l; the increase of the line broadening with increasing l is more rapid at larger values of (h2 + hk + k2).
2.4.
Stacking Faults
Another source of the anisotropic XRD line broadening in h-BN is a random stacking faulting of the {00l} planes. According to Warren [30], the randomly distributed stacking faults cause a fragmentation of coherently scattering domains (crystallites) in the 〈001〉 direction, thus for X-ray diffraction the crystallites appear smaller in the 〈001〉 direction than in the 〈hk0〉 directions. The XRD line broadening from these sf stacking faults, βhkl, increases linearly with increasing stacking fault probability αsf: βsfhkl ¼ 3
lα sf : c2 dhkl
ð13Þ
In contrast to the turbostratic defects, cf. Eq. (6) and Fig. 1a, only the widths of the diffraction lines with h − k = 3n ± 1 are affected by the randomly distributed stacking faults. The diffraction lines with h − k = 3n are not affected by these stacking faults. However, they can still be broadened because of a small crystallite size. The line broadening as calculated for a = 2.5044 Å and c = 6.6562 Å [18], and for αsf = 0.015 is displayed in Fig. 1d.
3. Identification and Quantification of the Microstructure Defects The different dependencies of the XRD line broadening on the diffraction indices hkl are summarised in Eqs. (6), (9), (12), & (13) and shown in Fig. 1. They indicate that the microstructure defects in hexagonal BN can be distinguished from each other when a sufficient number of diffraction lines is measured and analysed. The turbostratic disorder broadens mainly the diffraction lines hkl with high l; the diffraction lines hk0 and 00l are not affected. The puckering broadens mainly the diffraction lines 00l; the diffraction lines hk0 are unaffected. The edge dislocations {001}〈110〉 broaden in principle all diffraction lines hkl with higher l. However, the broadening of the diffraction lines 00l is negligible, as the Burgers vectors of these dislocations are perpendicular to the normal direction of the {001} lattice planes. The stacking faults broaden only the diffraction lines with h − k = 3n ± 1; other diffraction lines are unaffected. Turbostratic disorder and stacking faults are distinguishable on the basis of their different line broadening effects on the 11l lines. In Fig. 1, the line broadening caused by the individual microstructure defects is highlighted by solid lines for constant diffraction indices h and k, and by dashed lines for constant l. The trends observed for individual microstructure defects are unique. This allows the amount of turbostratic disorder, the magnitude of puckering, the dislocation density and the density of stacking faults to be quantified simultaneously. As the XRD line broadening depends linearly on the parameters that quantify the amount of the respective microstructure defects, i.e., on γ (1 − γ)−½, |Δc|, Mρ½ and αsf, see Eqs. (6), (9), (12) and (13), the total integral XRD line broadening can be approximated by a linear combination of the individual contributions. In this linear combination, the above parameters
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have the meaning of the weighting coefficients. Consequently, these parameters can be obtained from the measured (physical) line broadening by solving the equation MTij Mij P j ¼ MTij Bi
ð14Þ
which is equivalent to the least-squares refinement of the unknown microstructure parameters (Pj) from the physical line broadening (Bi). The matrix Mij is composed of column vectors that represent the contributions of the individual microstructure defects to the XRD line broadening. The rows of the matrix Mij stand for individual hkl reflections. The elements of Mij with respect to the different defect types (j = 1…4) are 2l Mi1 ¼ 2 πc dhkl
The upper index T in Eq. (14) means the transposed matrix. Pj is a column vector containing the refined parameters described above in the form: Pj ¼
γ pffiffiffiffiffiffiffiffiffi jΔcj 1−γ
pffiffiffi M ρ α sf
1 D
T :
ð20Þ
D in Eq. (20) is the crystallite size. The column vector Bi in Eq. (14) contains the measured integral breadths of the diffraction lines hkl that were corrected for the instrumental line broadening. The quality of the least-square fit [Eq. (15)] was judged by using the following goodness of fit:
ð15Þ
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 uX exp calc u u i βi −βi t Rw ¼ 100% X exp 2 βi i
ð16Þ
= MijPj. The estimated standard deviations of the with βcalc i refined microstructure parameters were calculated from the diagonal elements of the inverse matrix (MTijMij)−1,
ð22Þ
for turbostratic disorder, 2
l Mi2 ¼ 3 c dhkl for puckering defects,
h i 91 8 2 2 2 2 2 rffiffiffi < 4 3:2618 h þ hk þ k þ 1:7 10−4 l l = π b 0:02341 þ Mi3 ¼ 4 ; 2 : 9a2 c2 d
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u X exp calc 2 u
−1 u βi −βi i : σ j ¼ t MTij Mij n−p jj
ð23Þ
hkl
ð17Þ Here, n and p are the number of experimentally determined line widths and the number of refined parameters, respectively.
for the {001}〈110〉 edge dislocations, and Mi4 ¼
3l for h–k ¼ 3n 1 c2 dhkl
ð18Þ
Mi4 ¼ 0 for h–k ¼ 3n for stacking faults on basal planes. The last column in the Mij matrix is devoted to a constant XRD line broadening due to the small crystallites (according to Scherrer [31]) and reads Mi5 ¼ 1:
ð19Þ
4.
Materials and Methods
The routine described above was tested on samples of graphitic boron nitride containing the microstructure defects discussed above in different concentrations. Hot-pressed h-BN (HDBN) was purchased from HENZE Boron Nitride Products (HeBoSint®) in rod shape. It is characterised by a low
Fig. 2 – X-ray diffraction patterns of different h-BN samples: hot-pressed high-density boron nitride (HDBN), pyrolytic boron nitride (pBN) and chemically vapour deposited boron nitride (CVD-BN). Open circles represent the measured data, and the solid lines the result of the line profile fitting. The intensities are plotted in logarithmic scale. Individual XRD patterns were mutually shifted for clarity. In regions with significant overlap of heavily broadened diffraction maxima, individual peaks are indicated by dashed lines. The diffraction indices are displayed at the top of the graph.
M A TE RI A L S C HA RACT ER I ZA TI O N 86 ( 20 1 3 ) 1 9 0–1 9 9
concentration of microstructure defects and a good crystallinity. Pyrolytic h-BN (pBN) was purchased from Morgan Technical Ceramics. This material is produced in a high temperature CVD process and has a high density of puckering defects. Another CVD h-BN (CVD-BN), which was kindly provided by Dr. I. Petrusha (Institute of Superhard Materials at the National Academy of Sciences of Ukraine in Kiev), has been chemically vapour deposited at a low deposition temperature. This material is sometimes referred to as “turbostratic BN” because it has a very high density of turbostratic defects. All investigated materials fulfil the conditions for powder diffraction. The X-ray diffraction experiments were performed in Bragg– Brentano geometry using a Seifert FPM URD6.5 diffractometer that was equipped with a sealed X-ray tube with Cu anode and with a graphite monochromator located in the diffracted beam. The instrumental line broadening was determined using LaB6 purchased from NIST (SRM660b) [32]. Interpolation of the instrumental line broadening function was done with the equation postulated by Louër & Langford [33]. The experimental integral breadths were extracted from the measured diffraction patterns by using line profile fitting. Individual diffraction lines were approximated by an asymmetric Pearson VII function [34]. The physical line broadening was obtained by subtracting the instrumental line broadening from the measured integral breadths as usual for the Cauchy (Lorentz) shape of the diffraction lines [35]. The results obtained from XRD, in particular the crystallite sizes and the kind of the dominant microstructure defects, were verified by TEM and/or high resolution TEM (HRTEM). The (HR) TEM experiments were performed using a JEM 2200FS field emission TEM (Jeol), which is equipped with a Cs corrector in the primary beam (Ceos). The transmission electron microscope was operated at 200 kV acceleration voltage. TEM preparation was done by carefully cutting slices from the bulk starting materials. These were ground and mechanically polished. The final preparation step was the Ar+ ion thinning of the samples.
5.
identified that the stacking faults on the {001} planes also contribute to the XRD line broadening. The parameters obtained from the refinement are M2ρ = (6.6 ± 0.4) × 1012 m−2 and αsf = (4 ± 1) × 10−4. Using the first approximation of the pffiffiffiffiffi−1 Wilkens factor, M≈ ln b ρ0 ≈7 [23], the true dislocation density in HDBN was estimated as ρ ≈ 1011 m−2. The amount of the puckering was very low, Δc = (0.2 ± 0.1) × 10−3 Å. The degree of the turbostratic disorder was equal to zero within the accuracy of the least-square fit [Eq. (14)] and therefore neglected. This confirms the results of other authors, who did not observe turbostratic disorder in h-BN after hot-isostatic pressing [36]. The absence of the turbostratic disorder in HDBN can also be concluded from the absence of the asymmetry of the line shape of the h00 and hk0 diffraction lines (Fig. 2).
Results
The XRD patterns of h-BN with different densities of microstructure defects are shown in Fig. 2. HDBN with the lowest defect density produces the narrowest but still anisotropically broadened diffraction lines. The high density of puckering defects in pBN is apparent from the large broadening of the diffraction lines 00l. The high density of turbostratic defects in CVD-BN causes a rapid increase of the broadening of the diffraction lines 10l and 11l with increasing l. Therefore, only the 10 and 11 bands are visible. This phenomenon is illustrated on the diffraction lines 100, 101, 102 and 103 (dashed lines in Fig. 2). In contrast, the diffraction lines 00l are well pronounced, as they are not affected by the turbostratic defects (see Section 2.1).
5.1.
195
Hot-pressed High-density Boron Nitride (HDBN)
The large broadening of the diffraction lines 10l and a moderate broadening of the diffraction lines 00l (Fig. 3a) suggest that the edge dislocations {001}〈110〉 are the most dominant microstructure defects in HDBN (cf. Fig. 1c). The analysis of the microstructure defects with the aid of Eq. (14)
Fig. 3 – Dependence of the integral line widths on the magnitude of the diffraction vector as obtained from the line profile fitting (circles) and as calculated from the microstructure parameters determined using Eq. (14) (large crosses). Small crosses mark the line widths calculated for overlapping, very weak or extremely broadened diffraction lines, whose experimental counterparts were inaccessible. The line oscillating around 0 shows the differences between experimental and calculated data. The weighted reliability factor (Rw) was determined using Eq. (22).
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The constant part of the line broadening was quite low. This indicates a large crystallite size, which was calculated to be D = (260 ± 15) nm. This crystallite size was confirmed by TEM. The dark-field TEM micrograph (Fig. 4a) shows exemplarily a grain with the size between 250 and 350 nm. Also the dislocations were proven to be the most frequent microstructure defects in HDBN. In Fig. 4b, typical dislocation structures can be seen in the dark-field image that was recorded at the 224 diffraction spot in order to enhance the contrast of the dislocations with the Burgers vector 〈110〉.
5.2.
Pyrolytic Boron Nitride (pBN)
The anisotropy of the line broadening observed in pBN (Fig. 3b) is characteristic for graphitic boron nitride containing puckered basal planes (cf. Fig. 1b). Accordingly, a large magnitude of puckering, |Δc| = (0.19 ± 0.02) Å, has been determined with the aid of Eq. (14). Further microstructure defect that significantly contributes to the line broadening in pBN is the turbostratic disorder with the magnitude of γ = 0.14 ± 0.01. The crystallites in this materials are small, D = (14 ± 2) nm. There was no noteworthy contribution of dislocations and
stacking faults to the observed line broadening, thus the respective refinable parameters of these microstructure defects were regarded to be zero within the accuracy of this approach. HRTEM (Fig. 5) recognises the puckering in pBN as a waviness of the lattice planes {001} and as a variation of the interplanar spacing d00l. These phenomena are visible in the Fast Fourier Transform (FFT) of the HRTEM micrograph (inset in Fig. 5) as well. The waviness of the lattice planes {001} causes the azimuthal broadening of the diffraction spots (00l) the variation of the interplanar spacing d00l, their radial broadening. Due to the considerable turbostratic disorder, only the lattice planes {001} but no pronounced electron density modulations within these lattice planes are visible. Furthermore, HRTEM verified a small size of coherently scattering domains (crystallites) that was concluded from the XRD line broadening. It is worth noting that the crystallite size determined by XRD (D ≈ 14 nm) should be compared with the crystallite size obtained by HRTEM in the [hk0] directions. In these directions, mainly the disturbances marked by dashed lines in Fig. 5 destroy the coherence between the neighbouring domains. The size of the continuous regions ranges between 10 and 20 nm; this agrees very well with the crystallite size obtained from the XRD line broadening. In the [001] direction, the condition for coherent scattering between the subsequent {00l} lattice planes is mainly disturbed by the puckering, which is also the main source of the broadening of the XRD lines (00l), cf. Fig. 3b.
5.3.
Chemical Vapour Deposited Boron Nitride (CVD-BN)
In boron nitride that was chemically vapour deposited at low temperatures (CVD-BN), the turbostratic disorder, puckering and small crystallite size have been identified as the main contributions to the line broadening. The fitting of the integral
Fig. 4 – (a) Two partly overlapping h-BN grains in HDBN as seen using dark-field TEM. Their size agrees well with the mean crystallite size determined using XRD (approximately 260 nm). (b) Typical dislocation arrangement observed in the dark-field contrast. This TEM micrograph was taken using theh 224 i diffraction spot. The direction of the primary beam is 111 .
Fig. 5 – HRTEM micrograph of pBN showing the puckering (waviness) of the basal layers (lattice planes {001}). Regions with (apparently) continuous basal layers are separated by highly disturbed regions, where the basal layers appear discontinuous (dashed lines). The FFT of the HRTEM image is shown in the inset.
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6.
Discussion
6.1. Reliability of the Defect Densities Determined Using the Presented Method
Fig. 6 – HRTEM micrograph of CVD-BN and FFT/HRTEM (inset) illustrates a very small crystallite size (D ≈ 5 nm) and a locally wide orientation distribution of the 〈001〉 directions. The dashed lines mark the lattice planes {001}.
breadths from Fig. 3c using the microstructure model from Section 3 revealed the amount of turbostratic disorder γ = 0.73 ± 0.02, the puckering |Δc| = (0.12 ± 0.05) Å and the crystallite size D = (5 ± 1) nm. Similar to pBN, the contribution of stacking faults and dislocations to the overall line broadening observed in CVD-BN is too small to be reasonably quantified from the experimental data shown in Fig. 3c. The small crystallite size concluded from the XRD line broadening was confirmed by HRTEM, which revealed the grain size of approximately 5 nm (Fig. 6). Obviously, CVD-BN has a lower {001} local preferred orientation of crystallites than pBN. This can be seen mainly from different azimuthal broadenings of the diffraction spot (002), which are wider in CVD-BN than in pBN (compare FFT of the HRTEM micrographs in Figs. 5 and 6). The different local orientations of crystallites in CVD-BN and pBN are also indicated by the intensity ratios I(002)/I(100) and I(002)/I(110) that are higher in pBN than in CVD-BN (see Fig. 2). Because of the smaller amount of puckering in CVD-BN as compared with pBN, the interplanar distances d00l vary less in CVD-BN than in pBN. Both XRD and FFT/HRTEM recognise the smaller variance of d00l in CVD-BN as a smaller (radial) broadening of the diffraction lines 00l, see Figs. 3, 5 and 6. Like in pBN, the very high amount of the turbostratic disorder in CVD-BN makes the visualisation of the electron density modulations using HRTEM within the {00l} lattice planes impossible.
As already discussed above, transmission electron microscopy confirmed the presence of the dominant microstructure defects that was concluded from the XRD measurements. In order to be able to judge the reliability of the defect densities calculated from the XRD line broadening, the defect densities were additionally determined using alternative approaches based on the analysis of XRD data. Since the estimation of the amount of turbostratic disorder from the XRD line broadening has the most alternative approaches available, these were used to verify the amount of turbostratic disorder (γ) obtained from Eq. (14). The results of the comparison are collected in Table 3. The first approach was developed by Kurdyumov et al. [18], who found a quadratic dependence of the amount of turbostratic disorder in h-BN on the interplanar spacing d002. In our samples, the interplanar spacing d002 increased from HDBN over pBN to CVD-BN (see Table 3). Consequently, the degree of the turbostratic disorder calculated using the approach from Ref. [18], which is referred to as method #2 in Table 3, was γ ≈ 0, γ ≈ 0.4 and γ ≈ 1 for HDBN, pBN and CVD-BN, respectively. This trend corresponds to the increase of the turbostratic disorder found by applying our method (method #1 in Table 3), but the turbostratic disorder calculated from d002 is clearly overestimated. A possible reason is the effect of puckering on the interplanar spacing d002 that is not considered in Kurdyumov's model [18]. Another alternative approach [12,18] (method #3 in Table 3) determines the degree of the turbostratic disorder from the broadening of the XRD line 112, i.e. from β112. The relationship between β112 and the amount of turbostratic disorder is actually based on Eq. (6). In order to take the instrumental line broadening and a small crystallite size into account, Kurdyumov et al. [18] suggested correcting β112 with the aid of β110. As it can be seen from Fig. 1, the diffraction line 112 is heavily broadened by turbostratic defects and edge dislocations {001}〈110〉, but it is almost not affected by stacking faults and puckering, respectively. On the other hand, apart from the instrumental line broadening and small crystallite size, the diffraction line 110 is only broadened by dislocations. This approach revealed the amount of turbostratic disorder of γ = 0.03 for HDBN, γ = 0.15 for pBN and γ = 0.71 for CVD-BN, which are values being very close to those obtained from the refinement of the β(q) plots (Fig. 3).
Table 3 – Comparison of the amounts of the turbostratic disorder (γ) calculated using our approach (method #1), using the quadratic relationship between the amount of turbostratic disorder in h-BN and the interplanar spacing d002 [18] (method #2), and using the Kurdyumov method (#3) from Refs. [12] and [18], which is briefly described in Section 6.1. Method #1 Sample HDBN pBN CVD-BN
γ 0 0.14 ± 0.01 0.73 ± 0.02
Method #2
Method #3 −1
c = 2 · d002[Å]
γ
β110[Å ]
β112[Å− 1]
γ
6.657 ± 0.001 6.696 ± 0.003 6.883 ± 0.003
≈0 ≈0.4 ≈1.0
1.6 × 10−3 8.8 × 10−3 2.5 × 10−2
2.5 × 10−3 1.5 × 10−2 7.2 × 10−2
0.03 0.15 0.71
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If dislocations are the dominant microstructure defects, the calculation of the dislocation density from the line broadening is relatively straightforward (see Section 2.3). The procedure based on Eqs. (10) to (12) was successfully used by Klimanek & Kužel [26] and Dragomir & Ungár [28] to obtain the dislocation densities in cold-worked zirconium and titanium, respectively. The main drawback of this method is that it reveals only the dislocation density multiplied by the squared Wilkens factor, i.e., M2ρ. The Wilkens factor (M) describing the interaction of the dislocation field is required if the dislocation density is to be determined independently.
6.2.
Presented Approach Versus Single-stage Routines
As explained above, the presented routine handles the XRD data in two steps. In the first step, the measured diffracted intensities are approximated by analytical functions in order to separate the individual diffraction lines from each other and to obtain their integral breadths. In the second step, the dependence of the physical integral breadths on diffraction indices (after a correction for instrumental line broadening) is analysed using Eq. (14) in order to obtain the amounts of individual microstructure defects. Nowadays, the XRD patterns are frequently processed by using single-stage routines that allow the structure and microstructure parameters to be obtained in one step. Typical representatives of such routines employ the Rietveld approach [37,38], which refines the parameters of a structure model by fitting the calculated XRD patterns to the entire measured data. In materials containing microstructure defects, the crystallographic anisotropy of the XRD line broadening was always a factor, which deteriorates the agreement between the measured and calculated XRD patterns and which consequently influences the quality of the refined structure model. Therefore, already the earlier Rietveld-like computer programmes have handled the dependence of the XRD line broadening on the diffraction indices. Many of them employ the 4th order crystallographic invariants of diffraction indices as suggested by Popa [39] and Stephens [40]. The 4th order invariants are capable of describing the anisotropy of the crystal lattice deformation caused by the strain fields of microstructure defects. Thus, the anisotropic XRD line broadening calculated using these invariants improves the quality of the Rietveld fit in many cases. However, this method provides no information about the kind and density of the microstructure defects. Later on, physical models describing quantitatively the amount of the XRD line broadening for various microstructure defects were implemented into the Rietveld routines [17,41–45]. This allowed selected microstructure features like mean crystallite size and crystallite size distribution, dislocation density, stacking fault probability and degree of the turbostratic disorder to be quantified together with the lattice parameters and preferred orientation of crystallites. However, none of these routines is capable of modelling concurrently all microstructure defects observed in h-BN, i.e., turbostratic disorder, puckering, dislocations and stacking faults. The first reason is a missing description of the XRD line broadening caused by puckering. Secondly, strong correlations between individual microstructure parameters are expected, especially when several types of microstructure defects are present in the
samples simultaneously or if the number of refined microstructure parameters is large. In this context, it is worthy of noting that the Rietveld-like routines compare whole measured and calculated XRD patterns. Thus, these methods require some additional parameters describing the line shape, thereby they increase the number of free parameters further. From this point of view, the two-stage routine based on the presented approach is much more stable than the one-stage Rietveld-like routines, especially if several concurrent microstructure defects are present in the material under study and if their contributions to the total XRD line broadening are to be quantified at the same time. On the other hand, the Rietveld-based methods provide more detailed microstructure information once the dominant kind of the microstructure defects is known and refinement constraints can be applied.
7.
Conclusions
It was shown that the most frequent microstructure defects in hexagonal boron nitride, namely turbostratic disorder, puckering, dislocations and stacking faults can be distinguished from each other according to their effects on the amount and crystallographic anisotropy of the X-ray diffraction line broadening. A routine was suggested that allows both the dominant kind of the microstructure defects to be identified and quantified by suitable parameters. For this purpose, the measured XRD line broadening is regarded as a linear combination of the unique XRD line broadening from specific microstructure defects. For the quantification of the turbostratic disorder, dislocations and stacking faults, the existing models were employed. For the quantitative description of the (uncorrelated) puckering of the basal layers, a new model was created, which assumes that the puckering leads to a variation of the lattice parameter c. The routine was tested on hot-isostatically pressed h-BN, pyrolytic h-BN and low-temperature CVD h-BN. In the hot-isostatically pressed h-BN, the dominant microstructure defects are the edge dislocations {001}〈110〉 and the stacking faults on the {001} planes; the amount of puckering is very low and the turbostratic defects are almost absent. The microstructures of the pyrolytic h-BN and low-temperature CVD h-BN are dominated by the puckering and by the presence of turbostratic defects. The puckering is the most important microstructure defect in pyrolytic h-BN, whereas the turbostratic disorder is the typical microstructure feature in CVD h-BN.
Acknowledgements This investigation was carried out in the frame of the Freiberg High Pressure Research Centre (FHP) financed through the Dr. Erich Krüger Foundation of the TU Bergakademie Freiberg. The TEM JEOL 2200 FS was purchased from the funds of the Cluster of Excellence “Structure Design of Novel High-Performance Materials via Atomic Design and Defect Engineering (ADDE)” which is supported by the European Union (European Regional Development Fund) and the Ministry of Science and Art of Saxony (SMWK). The authors thank Dr. Zdeněk Matěj (Charles University Prague) for valuable discussion.
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