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Crystal thickness and extinction distance determination using energy filtered CBED pattern intensity measurement and dynamical diffraction theory fitting
Ultramicroscopy 87 (2001) 5–18
Crystal thickness and extinction distance determination using energy filtered CBED pattern intensity measurement and dynamical diffraction theory fitting D. Delillea,*, R. Pantelb, E. Van Cappellenc a
FEI France, 22 avenue Descartes, BP 45, F-94454 Limeil-Brevannes, France b ST Microelectronics, 850 avenue Jean Monnet, F-38926 Crolles, France c FEI Company, 7451 N.W. Evergreen Parkway, Hillsboro, OR 97124-5830, USA Received 27 March 2000; received in revised form 8 August 2000
Abstract A new method for measuring thickness and extinction distance of single crystals based on computed adjustment of measured and calculated CBED pattern intensity profiles is presented and discussed. The experimental beam intensity distribution is measured from an energy filtered CBED pattern recorded on a CCD camera. The calculated profile is based on dynamical diffraction theory, and with the two-beam approximation the analytical expression contains only two free parameters: specimen thickness t and extinction distance xg . Parameter refinement through minimization of the difference between experimental and calculated intensity profiles is carried out using OriginTM 5.0 software from Microcal. The iterative procedure always converges to a unique solution in a few seconds, yielding an accurate value for both thickness and extinction distance. The method is extensively tested on silicon using the (0 0 4) Bragg reflection. On specimens in the usual TEM thickness range, the method gives result similar to the conventional (P.M. Kelly et al., Phys. Stat. Sol. A31 (1975) 771; S.M. Allen, Philos. Mag. A 43 (1981) 325) graphical methods, both based on the measurement of fringe spacing. Moreover, it is shown that the calculation matches perfectly both the positions of the minimums and maximums as well as the amplitude of maximums. For any single intensity profile, specimen thickness and extinction distance can be determined with a precision of about 0.2%. A statistical comparison of our method with the Kelly and Allen techniques, based on more than 50 experiments, shows an improvement in measured extinction distance dispersion. Using 197 keV electrons, and liquid-nitrogen cryo-holder, the new technique yields an experimental value of 161 3 nm for the extinction distance for silicon with the (0 0 4) Bragg reflection. The equivalent tabulated value at 0 K is about 156 nm. Using the Kelly and Allen methods, the extinction distance is found to be 162 6 nm. The improvement in precision is a direct consequence of matching the intensity profile envelope, which contains information on the extinction distance. Also the accuracy of thickness determination is improved and is around 0.5 to 1% for common specimen thickness. The minimum measurable sample thickness is shown to be two to three times thinner than with the Kelly and Allen methods (0.3xg as opposed to 0.8xg ). With no independent calculation of the extinction distance needed, the method is also applicable on unknown crystals. The method is fast, simple and can be easily automated. # 2001 Elsevier Science B.V. All rights reserved. Keywords: CBED; Energy filtering; Extinction distance; Thickness measurement
*Corresponding author. Tel.: +33-4-76-76-45-84; fax: +33-4-76-76-45-99. E-mail address: [email protected] (D. Delille). 0304-3991/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 0 ) 0 0 0 6 7 - X
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1. Introduction
Maximums : tanðpDktÞ ¼ pDkt;
Accurate specimen thickness measurements are needed for a wide variety of well established TEM applications ranging from image simulations (HRTEM) for validation of proposed structure models to correction schemes for absorption and fluorescence effects in quantitative EDX analysis and beam broadening calculations to determine the attainable analytical spatial resolution. Direct sample thickness measurements can be carried out using projected interface observations at different angles. Electron energy loss spectroscopy (EELS) can also yield specimen thickness [1], provided that the mean free path for inelastic scattering can be determined independently. Convergent beam electron diffraction (CBED) as a means to measure crystal thickness was used as early as 1940. MacGillavry [2] used two beam dynamical diffraction theory to link the fringe minimums observed in CBED patterns and first described by Kossel and Mo¨llensted [3], to the specimen thickness:
defining a set of Dkt ¼ xk0 :
ðs2i þ 1=x2g Þt2 ¼ n2i ;
ð1Þ
where si is the deviation of the ith minimum from the exact Bragg position, xg is the extinction distance and ni an integer number. Ackermann [4] showed that the specimen thickness t could be obtained from the slope of a plot of s2i versus n2i . Kelly et al. [5] first and Allen [6] later, described a more elaborate plotting technique for thickness determination. In the two beam approximation, curve for the diffracted beam intensity the rocking Fg 2 is given by [7]: 2 Fg ¼ sin2 b sin2 ðpDktÞ; ð2Þ b ¼ tan1 ð1=sxg Þ;
ð3Þ
Dk ¼ ½ð1 þ ðsxg Þ2 1=2 x1 g
ð4Þ
By differentiation with respect to s, the expression for the minimums and maximums of 2 Fg can be found Minimums : Dkt ¼ integer; defining a set of Dkt ¼ nk
ð5Þ
ð6Þ
Kelly et al. [5] expressed Eq. (5) as ðsi =nk Þ2 ¼ ð1=xg Þ2 ð1=nk Þ2 þ ð1=tÞ2 :
ð7Þ
Allen (1981) expressed Eq. (6) as ðsi =xk0 Þ2 ¼ ð1=xg Þ2 ð1=xk0 Þ2 þ ð1=tÞ2 :
ð8Þ 2
Kelly et al. [5] proposed a plot of ðsi =nk Þ versus ð1=nk Þ2 and Allen [6] a plot of ðsi =xk0 Þ2 versus ð1=xk0 Þ2 . Eq. (7) and (8) do plot the same straight line characterized by a slope equal to ð1=xg Þ2 and an intersection with the Y-axis (s ¼ 0) equal to ð1=tÞ2 . By measuring the slope and the intersection with the Y-axis, the extinction distance and the sample thickness are derived. However, both methods (Kelly and Allen) are subject to errors when wrong sets of nk or xk0 are chosen to match thickness and extinction distance. Some graphical solutions yield too large values for xg , and too small values for t, this happens when the curve is slightly (usually unnoticeably) curved with a negative second derivative (concave with respect to the X-axis). Inversely too small values for xg , and too large values for t will result when the second derivative is positive. The right solution is a straight line giving the correct value for the effective extinction distance and the specimen thickness. However, Allen has shown that multiple beam diffraction effects can induce distortions and that anomalous absorption displaces minimums and maximums differently. Particular experimental conditions can lead to a difficult choice for the right solution using the Kelly and Allen graphical plot, for example when a low number of minimums and maximums are used and when their positions are measured with low accuracy. By knowing the extinction distance the choice for the best set of nk or xk0 is made easier. The extinction distance can be calculated independently using the expression xg ¼ pVc cos yB =lFg ; where Vc is the volume of a unit cell, and Fg is the structure factor FðyB Þ for the g reflection. Calculating Fg necessitates the atomic scattering factors f ðyÞ, which are tabulated in the International
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Crystallographic Tables III [8]. When these data are not easily available, or if the observed material is unknown, the extinction distance cannot be calculated. Under such conditions, a doubt may subsist when choosing a plotting set for a Kelly or Allen graph. The inherent limitation of the Kelly and Allen methods is that the information carried by the intensity is not used. Only the fringe positions are measured and a limited number of points are considered, i.e. the maximums and minimums. The diffracted beam intensity exhibits oscillations convoluted a damping envelope. The with 2 expression for Fg calculated with the two-beam approximation dynamical diffraction theory (Eq. (2)), describes these oscillations. The ½1 þ ðsxg Þ2 1 term describes the envelope and the full-width at half-maximum of the envelope equals 2=xg . Obviously, intensity measurement will enhance and complement the information contained in the fringe spacing. The intensity distribution specifically contains valuable information about the extinction distance. In this communication we propose a method to determine both specimen thickness and extinction distance, using a quantitative measurement of diffracted beam intensities combined with theoretical modeling. The tools used to measure the CBED pattern intensity distributions are now commonly available on modern TEMs (i.e. energy filter and CCD camera). A CCD camera exhibits a linear dynamic range of about 104 counts and an energy filter is used to eliminate the inelastically scattered electrons so as to improve the CBED patterns. An intensity profile is extracted from the diffraction pattern, using a two-beam CBED condition. The transmitted beam and diffracted beam are recorded simultaneously with 2a single exposure. Then the intensity profile Fg as function of the deviation from exact Bragg diffraction is compared with Eq. (2). Only two free parameters t and xg are used to refine experimental and calculated profiles. The data analysis software OriginTM 5.0, from Microcal Inc., running on a personal computer, is used to fit the experimental CBED profile, with a non-linear least-squares fit method. This method is based on a modified Levenberg–
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Marquardt algorithm [9], which is the most widely used among non-linear functions refinement methods. It allows to easily determine both parameters. The method is evaluated on a silicon crystal, using the (0 0 4) Bragg reflection and examples of typical results are presented in order to illustrate the real potential and advantages of this technique. A study is made as function of specimen thickness, down to 30% of the extinction distance, and an extensive comparison between the Kelly and Allen methods and ours is carried out using the silicon (0 0 4) Bragg reflection and involving more than 50 experiments with varying sample thickness. Particular attention was paid to extinction distance determination, as this parameter is supposed to be constant for all experiments, and also because it can be calculated independently. This comparison gives a clear evaluation of the precision of thickness and extinction distance measurements for all methods.
2. Experimental details The experiments are carried out on a Philips CM200 FEG microscope, equipped with a gatan imaging filter (GIF). To avoid radiation damages and possible carbon contamination when a small probe is focused on the specimen, a cooling holder is used at liquid-nitrogen temperature. The nanoprobe mode is used to produce a small probe with a 4.5 mrad convergence semiangle. A silicon specimen exhibiting a very shallow wedge angle and prepared using a focused ion beam (FIB) [10] is oriented into a two-beam diffraction condition, defined by the (0 0 4) Bragg reflection. Multiple CBED experiments are carried out perpendicular to the wedge edge, in order to explore a large range of sample thickness. The diffraction camera length is chosen so as to fit both the transmitted and the diffracted discs on the same image. Inelastically scattered electrons are eliminated using an energy slit of 10 eV centered around 197 keV elastic energy. Quantitative CBED patterns are acquired on the 1 K 1 K CCD camera of the GIF and an intensity profile perpendicular to the fringes is extracted from each CCD image, using Digital MicrographTM 2.5
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software. Pixel averaging, parallel to the fringes, is used to improve the signal-to-noise ratio. Finally, the profile inside the diffracted disc is plotted versus the deviation from the exact Bragg conditions, calibrated, with respect to the Bragg angle 2yB between the transmitted and the diffracted disc centers. Therefore, the experimental profile represents 2 Fg ðsÞ with s ¼ ðDy=2yB Þðl=d 2 Þ: hkl This experimental profile containing typically 300 measured intensities (i.e. 300 pixels) is compared with calculation using the analytical expression of Eq. (2). The best parameter solution (extinction distance xg and specimen thickness t) is determined using OriginTM software running under Windows NTTM and the iteration uses a non-linear Levenberg–Marquardt least-squares fit algorithm. All measured intensities have the same weight in the least-squares-iteration. Notwithstanding the fact that the intensity profiles are acquired using an energy filter, intensities corresponding to the minimums are never reduced to zero, and this is in contradiction with the dynamical theory of electron diffraction. This phenomenon is due to several effects: multiple scattering in the bulk crystal, plasmon scattering, thermal and surface effects, artifacts related to amorphous layers left by the FIB preparation technique and possible surface contamination. In the non-linear
least-squares refinement, this background is modeled by the sum of a constant base line and a broad symmetrical Gaussian centered on s ¼ 0. The least-squares iterative calculation needs a few seconds to find a unique solution for xg and t, and it also yields a relative accuracy figure. The complete process can be automated with only limited operator input after the CBED pattern acquisition. The described method besides being more accurate is also easier and faster than the Kelly and Allen plot methods.
3. Experimental results and data processing To obtain a (0 0 4)/(0 0 0) two-beam condition in silicon, the crystal is roughly tilted by 108 starting from the h1 1 0i zone-axis orientation. The convergent electron beam diffraction patterns are zero-loss filtered to eliminate the inelastically scattered electrons (see Fig. 1). The convergence semi-angle is adjusted to 4.5 mrad, so as to maximize the disc diameters without overlapping the discs and the camera length is chosen to use a maximum number of pixels of the 1 K 1 K CCD camera. The exposure time is adjusted in order to reach saturation of the CCD camera in the transmitted beam (around 10 000 counts). A line profile across the two disc centers is generated using Digital MicrographTM
Fig. 1. Convergent beam electron diffraction pattern of a 190 nm thick silicon crystal, acquired on a CCD camera, using 197 keV electrons. Bright diagonal lines appearing on the right-hand side of the (0 0 4) disc make part of the (0 0 2) disc intensity, overlapping (0 0 0) and (0 0 4) diffraction discs.
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software. Transverse pixel averaging is used to improve statistics and increases the signal to noise ratio to about 103. Fig. 2 is the line profile intensity obtained from Fig. 1. This profile can be calibrated in inverse nanometers using the Bragg angle value between the central local maximum of the transmitted disc and the local minimum of the (0 0 4) disc. The deviation parameter s becomes 2 s ¼ ðDy=2yB Þðl=dhkl Þ:
Fig. 3 shows the intensity profile of (0 0 4) diffracted disc as a function of the deviation parameter. The width of the profile is about 300 pixels and a high signal-to-noise ratio is observed. These are the optimum conditions for an accurate comparison between calculated and measured intensities. Two-beam dynamical diffraction theory predicts zero intensity for the minimums, whereas the experimental curve shows 10–15% residual intensity under the minimums close to s ¼ 0. Energy filtering helps to eliminate inelastically scattered electrons that contribute to the experimental background. However, some residual background will remain because of anomalous absorption, non-perfect filtering (the slit is not infinitely small) and a camera point spread function that is far from the ideal delta function. Moreover, as described by Peng et al. [11] and simulated by Dudarev et al. [12], a part of this non-zero background is due to plasmon scattering.
Fig. 2. Profile obtained from Fig. 1, with an average of 50 pixels width across (0 0 0) and (0 0 4) diffraction discs.
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From the maximums and minimums of Fig. 3, a Kelly and Allen plot can be generated (see Fig. 4). Because of the optimum experimental conditions, these measurements are a clear test of the performance and limitations of the Kelly and Allen graphical method. Fig. 4 presents a Kelly and Allen plot of ðsi =nk Þ2 versus ð1=nk Þ2 and ðsi =xk0 Þ2 versus ð1=xk0 Þ2 , for two sets of nk and xk0 series. From these two sets of possible solutions, the best one is the arguably straightest line which gives a specimen thickness equal to 198.5 nm and an extinction distance equal to 159.6 nm.
Fig. 3. Intensity profile of the (0 0 4) diffraction disc, extracted from Fig. 2, and normalized versus the deviation from the exact Bragg reflection.
Fig. 4. Graphical Kelly and Allen graphical method applied to the profile of Fig. 3.
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The theoretical value for the extinction distance of the 0 0 4 Bragg reflection in silicon for an electron beam energy of 197 keV can be calculated using the following expression: xg ¼ pVc cos yB =lFg ; where Vc ¼ a3 ¼ 160:2 A˚3, y ¼ 0:538; Fg ¼ 8f ðyB Þ; f ðyB Þ ¼ 1:605 A˚, and l¼ 2:51 102 A˚. This yields a theoretical value for extinction distance equal to xg ¼ 156 nm. The experimentally determined xg value of 159.6 nm is in good agreement. The difference of around 2% can be due to the Debye–Waller factor ðexpðB2g =4ÞÞ, although small at liquid-nitrogen temperature and other experimental conditions affecting the extinction distance such as weak multiple beam effects and plasmon excitations. The next figure shows the new technique applied on the experimental intensity profile of Fig. 3. OriginTM 5.0 software is used to compare the experimental profile with a calculated one using Eq. (2) and the difference is minimized using t and xg as adjustment parameters. To cope with the non-zero intensity of the experimental minimums, a background is first subtracted using a large smooth function. As described in Fig. 5, this smoothing function consists of a base line component and a unique Gaussian function. Sources of this background are of different nature. As mentioned above, silicon wedges are FIB prepared, and this technique is known to leave amorphous layers on both sides of the sample [13]. Diffraction of amorphous layers yields rings related to the local mean atomic distances. These rings are broad and have a low intensity overlapping the diffraction discs. This can contribute to the background measured in the diffracted disc, although no significant changes were observed in the background between samples of variable amorphous surface layers (obtained using different FIB acceleration voltages [13]). Another source of this background can be the convolution of the real sample transfer function (diffraction plus anomalous interactions) with an instrumental transfer function. The latter one contains the point-spread function of the CCD camera, which can be neglected while considering the pixel number used in each line scan. Moreover, the instrumental
transfer function has far less impact in diffraction mode compared to imaging mode. Main contribution to this background can be found in anomalous absorption, well described by Hirsch et al. [7] and modeled by Doyle and Turner [14], and Humphreys and Hirsch [15], and in plasmon scattering, first noted by Peng et al. [11]. Most of these plasmon interactions are filtered using the Gatan GIF, but, due to the finite size of the filter slit, plasmon contribution must be taken into account. Hirsch et al [7] described the case of an absorbing crystal using the following expression: 2 Fg ¼ sin2 bðsin2 ðpDktÞ þ sinh2 ðpDk0 tÞÞ
expð2pt=x00 Þ;
ð9Þ
where b ¼ tan1 ð1=sxg Þ;
Dk0 ¼ ð1 þ ðsxg Þ2 Þ1=2 x01 g ;
with x0g and x00 being anomalous absorption distances. The term expð2pt=x00 Þ describes the intensity attenuation caused by sample thickness and is independent of s. The second term sin2 b sinh2 ðpDk0 tÞ is symmetric around s ¼ 0, has a maximum at s ¼ 0 and tends to zero for large s values. This shape roughly corresponds with the background observed in all the experiments. However, as the hyperbolic sine is not very different from a Gaussian function, we have chosen, for the sake of simplicity, a Gaussian background model. Fig. 5 shows good agreement between the experimental and calculated intensities. A solution yielding a similar extinction distance and specimen thickness than those obtained from the Kelly and Allen graph (Fig. 4) not only reproduces the right positions of the minimums and maximums, but also fits perfectly the oscillation amplitudes. This confirms that the two-beam dynamical theory perfectly applies to the experimental conditions used to obtain the profile of Fig. 3, and that the background subtraction does not introduce too many artifacts. The other advantage of a full statistical analysis is the accuracy estimate. In this case, xg ¼ 160:06 0:36 nm and t ¼ 198:55 0:24 nm, which means that we have a relative precision of about 2 103
D. Delille et al. / Ultramicroscopy 87 (2001) 5–18
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Fig. 5. Two-beam dynamical diffraction calculated profile, and comparison with experimental CBED profile of Fig. 3. Gaussian and baseline extraction of the background are presented.
Fig. 6. Comparison between calculated and experimental CBED profiles, using fixed values of respectively 157 and 163 nm for xg (systematic 2% error around 160 nm), and 198.55 nm for t.
for the extinction distance and 103 for the specimen thickness. The envelope of oscillations in a two-beam condition is related to the extinction distance, and the width at half maximum of the envelope is equal to 2=xg . Therefore, the solution for xg should be accurate and unique, in order to reproduce the right sequence of decreasing amplitude. This hypothesis can be tested by arbitrarily fixing the extinction distance within 2% of its best-estimated value and check whether a match is possible between experiment and theory. Fig. 6 shows fits using xg ¼ 157 nm and xg ¼ 163 nm, both with the same optimum sample thickness of 198.5 nm. This clearly shows that a non-optimized xg cannot explain the right amplitudes of the maximums, and that the fitting process is sensitive to 0.2% variation of the extinction distance value. If both xg and t are varied, neither the amplitude of maximums nor the positions of maximums and minimums are accurately fitted. In this way, Zuo and Spence [16] noticed that under two-beam diffraction conditions, the intensity at s ¼ 0 was strongly dependent on xg variations, with a maximum sensitivity for an optimum thickness of t ¼ 1:25xg . In this range, a 1% variation in xg gives a 6% variation in intensity. Fig. 6 shows very similar results, with an intensity variation of about 11% for a 2% xg variation. This confirms a high relative precision
of 0.2% with which both xg and t are measured in Fig. 5. Figs. 7a–e are examples of CBED patterns acquired at different thickness on a silicon wedge, ranging from 2xg down to 0:3xg , and Figs. 8a–e are the corresponding experimental and calculated intensity profile fits together with the Kelly and Allen plots. Table 1 summarizes the comparison between graphical method and intensity fit method results. For specimen thickness range above 0:8xg the two methods give similar results with less than 2% difference. Nevertheless, Kelly and Allen methods give only a set of possible solutions, instead of the unique solution obtained using the intensity fit method. Below 0:8xg , the extinction distance determined by graphical Kelly and Allen plot becomes unstable (see dispersion in Fig. 8b(2)). Under 0:6xg , no acceptable solution is found by this way for xg and therefore no solution for thickness t. By comparison, an accuracy of 1% is still obtained on thickness determination below 0:3xg using the intensity fit method. Fig. 9a is a graph of the extinction distance as function of sample thickness using the Kelly and Allen plot method. The measured extinction distances are dispersed around a mean value, which tends to increase with decreasing thickness. Above a sample thickness of 130 nm, which corresponds to 0:8xg , the mean value is 162 10 nm and 70% of the data points are
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Fig. 7. (a)–(e). Series of 197 kV experimental CBED patterns, showing the (0 0 4) diffracted and (0 0 0) transmitted discs for different silicon crystal thickness.
D. Delille et al. / Ultramicroscopy 87 (2001) 5–18
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Fig. 8. a(1) Intensity profile fit corresponding to Fig. 7a: extinction distance xg ¼ 163:65 nm and thickness t ¼ 280:97 nm. a(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 281 nm thick silicon crystal. b(1) Intensity profile fit corresponding to Fig 7b: extinction distance xg ¼ 160:14 nm and thickness t ¼ 150:17 nm. b(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 150 nm thick silicon crystal. c(1) Intensity profile fit corresponding to Fig 7c: extinction distance xg ¼ 166:00 nm and thickness t ¼ 125:47 nm. c(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 125 nm thick silicon crystal. d(1) Intensity profile fit corresponding to Fig 7d: extinction distance xg ¼ 164:85 nm and thickness t ¼ 94:21 nm. d(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 95 nm thick silicon crystal. e(1) Intensity profile fit corresponding to Fig 7e: extinction distance xg ¼ 160:28 nm and thickness t ¼ 45:30 nm. e(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 45 nm thick silicon crystal.
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Fig. 8. Continued.
Table 1 Comparison between Kelly and Allen methods and ours, based on results of Fig. 8. Thickness range 2 xg xg 0:8 xg 0:6 xg 0:3 xg
This work
Kelly and Allen
Comments
t (nm)
xg (nm)
t (nm)
280.97 0.29 150.17 0.15 125.47 0.06 94.21 0.40 45.30 0.43
163.65 0.27 160.14 0.35 166.00 0.14 164.85 4.46 160 48
280 163 152 163 125 166 94 185 No coherent solution
concentrated around 162 6 nm. Under 130 nm thickness, the extinction distance starts to increase, and below 110 nm or 0:7xg the measured values become unacceptable. To conclude, the Kelly and Allen plot methods allow measuring the extinction
xg (nm) Very similar results for the two methods. Some dispersion in Kelly and Allen plot. xg starts to diverge using Kelly and Allen method. Thickness determination with 1% accuracy using intensity fit method.
distance with a 4% relative precision for samples thicker than 0:8xg . Fig. 9b handles the same data as Fig. 9a, but using the intensity fit method. Below 100 nm thickness the error bars are indicated. An increase
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Fig. 9. (a) Plot of extinction distance versus crystal thickness : each point represents one of the 50 silicon crystals analyzed using Kelly and Allen graphical method. (b) Plot of extinction distance versus crystal thickness : each point represents one of the 50 silicon crystals analyzed using our non-linear least-squares fit method. (c) Histograms extracted from graphs (a) and (b). The two dashed curves are the fitted Gaussian functions corresponding to each histogram. The xg step (i.e. bar width) is 1 nm.
in the extinction distance at low thickness observed while using the Kelly and Allen plot method is virtually non-existent. Above 130 nm, the mean value is 161 5 nm and most of the values (70%) are concentrated around 161 3 nm. Above 130 nm thickness or 0:8xg , the intensity fit method allows to measure the extinction distance
with a 2% relative precision. For 100 and 50 nm thickness the error drops to 3 and 6% respectively, without significantly changing the value. Not only does accuracy and precision of extinction distance and sample thickness measurements improve, when using the intensity fit method, but also the minimum usable sample thickness is decreased
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from 0:8xg to 0:3xg when compared to the Kelly and Allen technique. The histograms of Fig. 9c represent the extinction distances obtained by using both methods and only the most favorable sample thickness range between 130 and 230 nm was considered. This leaves a total of 36 measurements. The intensity fit method clearly yields a narrower distribution than the Kelly and Allen plot method. Statistical analysis of the histograms through Gaussian distribution fits confirms the better accuracy of the intensity fit technique and gives the following numbers: xg ¼ 161 3 nm for the new method and xg ¼ 162 6 nm for the conventional method. To conclude, the accuracy in extinction distance measurement is around 2% using the new intensity fit technique, provided that the sample thickness is above 0:8xg . With the Kelly and Allen method it is generally accepted that the error on the thickness measurements is less than on the extinction distance measurements. The same holds for the intensity fit technique and as such it is fair to say that the precision in thickness measurement is around 0.5– 1%.
4. General discussion In our simulation procedure, we only have considered the true two-beam conditions to simulate the theoretical intensity profiles. Considering multiple beam effects necessitates the use of the matrix formulation of Bloch waves intensity [7,16–19], and the anomalous absorption modeling involves the use of complex atomic potentials, leading to the use of a large number of parameters. The aim of our work is to propose a thickness measurement method as simple as the Kelly and Allen ones, but more accurate and easy to use. As shown by Gjonnes et al. [19], we replace the Uð0 0 4Þ coefficient by its effective value, in order to fit experimental profiles. So, the extinction distance and structure factor values refined using our method are effective values, shown by Gjonnes et al. [19] to be very near absolute values using the two-beam approximation. So we suggest to choose the best two-beam orientations in order to limit
the refinement conditions around two key parameters: t and xg . In the case of the (0 0 4) reflection in a diamondlike crystal structure, the two-beam conditions are easily satisfied because the systematic Bragg reflections (0 0 2), (0 0 2) and (0 0 6) are not present on the diffraction pattern (structure factor equal to zero). The only possible reflections, (0 0 4) and (0 0 8) have a very high deviation parameter s, and their intensities are negligible. Experimental care are, essentially, sample cooling at liquid nitrogen temperature, reduced beam exposure time and minimization of the related carbonaceous contamination, and of course energy filtering. In such a condition, the background is reduced to its minimum value (10–15% of the total intensity), but is still present in the intensity profiles. In order to simplify our parameters refinement, we have modeled this background as a gaussian one instead of considering the anomalous absorption and plasmon scattering [11,12] and their heavy formalism. According to the experimental results presented above, our intensity fit method yields similar results as the graphical methods of Kelly and Allen, albeit with a better accuracy and precision and it is applicable to thinner samples. Kelly and Allen have estimated the accuracy of their thickness measurements to be around 2%, at least for a sample of reasonable thickness. They consider three minimums or maximums in the diffraction disc to be the lower limit of their technique. In this case they suggest that the extinction distance be measured in a thicker part of the specimen and to use this value in the plot. While using our intensity fit technique, the OriginTM 5.0 software matching results for a particular experiment shows a relative accuracy of about 0.1–0.2% for both the specimen thickness and extinction distance on the condition that the sample thickness is in the range between 0:8xg and a few xg . However, when a large number of experiments are carried out (Fig. 9b), the extinction distance accuracy drops to around 2%. The origins of this dispersion are the cumulative effects of experimental errors. The major contributions are accuracy of xB measurement (0.2% 1 pixel per
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500 pixels), image distortion in the energy filter and CCD camera (less than 0.3%), errors due to background modeling (probably less than 0.5%) and variable beam-induced thermal effects (around 1%). The improvement in extinction distance accuracy using our intensity fit method, over the Kelly and Allen method, is a factor of two for specimen thickness in the range of 0:8 xg to few xg (2% instead of 4%). For thinner samples, the difference becomes even more significant as the intensity fit method can measure xg on samples down to 0:3 xg , whereas the Kelly and Allen technique becomes virtually useless below a thickness of 0.8 xg . A statistical analysis determined the extinction distance of the (0 0 4) diffracted beam in silicon at 197 keV to be 161 nm 2% (Fig. 9c), whereas the value calculated from tabulated atomic scattering factors equals 156 nm. The difference of 3% might be explained by thermal agitation described by the Debye–Waller factor expðB2g =4Þ, although small at liquid nitrogen temperature. Anomalous absorption and multiple beam effects also influence the effective extinction distance. The same extinction distance measured using a Kelly and Allen graph yields 162 nm and the standard deviation is 4%. The accuracy with which sample thickness can be measured is around 0.5 to 1% for intensity fit technique even for very thin samples (0:3xg ), whereas the Kelly and Allen method fails for specimen is of thickness below 0:6xg . The Kelly and Allen plot techniques limit the information extracted from the diffracted disc to the fringe positions (the positions of minimums and maximums). The intensity information is lost. The two-beam dynamical diffraction approximation allows, besides predicting the positions of the minimums and maximums, to use the intensity information. The envelope of intensity profile oscillations is directly related to the extinction distance since the full-width at halfmaximum equals 2=xg . Beyond the envelope, the rocking curve and the extinction distance are intricately connected and as such modeling the measured intensities yields accurate information about xg . Moreover, Zuo and Spence [16] have noticed that the intensity at exact Bragg angle
17
(i.e. s ¼ 0 in the intensity profile) is strongly xg sensitive. Fig. 6 shows the extreme sensitivity of the rocking curve to the extinction distance and for a particular intensity profile, a unique value with an accuracy of 0.2% can be found. This is a distinct advantage over the Kelly and Allen plot methods where wrong choices for the extinction distance can be made, which is particularly problematic when the extinction distance cannot be calculated from tabulated atomic scattering factors. This is especially true when dealing with unknown crystals and a wrong extinction distance will lead to a significant error in the thickness measurements. With the intensity fit method it is straightforward to measure extinction distance and thickness of an unknown crystal, as only the Bragg angle must be determined.
5. Conclusions A new procedure for mono-crystal specimen thickness and extinction distance measurements using convergent beam electron diffraction is presented. Energy filtered and digitally recorded CBED patterns are used to produce the experimental intensity profile (the so-called rocking curve) which is fitted using the non-linear leastsquares refinement of a simulated one. The calculated profile consists of a two-beam dynamical approximation rocking curve and a background to cope with the experimental conditions. The iterative procedure optimizes two fit parameters, the extinction distance and the specimen thickness, and always converges towards a unique solution in a few seconds. The match between experimental and predicted is extraordinary and the accuracy of thickness measurements is between 0.5 and 1%. The method also allows to process very thin samples not suited for the conventional techniques. The effective extinction distance is also measured with a high accuracy (2%). Recent developments such as CCD cameras and processing software have made the method both fast and simple, but further automation can still be envisaged. The method is also applicable on
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unknown crystals since no independent calculation of the extinction distance is needed.
References [1] R.F. Egerton, Electron Energy Loss Spectroscopy in the Electron Microscope, Plenum Press, New York, 1986. [2] C.H. MacGillavry, Physica 7 (1940) 329. [3] W. Kossel, G. Mo¨llensted, Ann. Phys. 5 (1939) 113. [4] I. Ackermann, Ann. Phys. 6 (1948) 19. [5] P.M. Kelly, A. Jostsons, R.G. Blake, J.G. Napier, Phys. Stat. Sol. A 31 (1975) 771. [6] S.M. Allen, Philos. Mag. A 43 (1981) 325. [7] P.B. Hirsch, A. Howie, R.B. Nicholson, D.W. Pashley, M.J. Whelan, Electron Microscopy of Thin Crystals, Butterworths, London, 1965. [8] International Crystallographic Tables III. [9] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes in C, 2nd edition, Cambridge University Press, Cambridge, 1992.
[10] R. Pantel, G. Auvert, G. Mascarin, Microelectronic Engineering 37/38 (1997) 49. [11] L.M. Peng, S.L. Dudarev, M.J. Whelan, Phys. Lett. A 175 (1993) 461. [12] S.L. Dudarev, L.M. Peng, M.J. Whelan, Phys. Rev. B 48 (1993) 13 408. [13] D. Delille, R. Pantel, G. Auvert, E. Van Cappellen, Microscopy and Microanalysis 1999 Proceedings, Springer, New York, 1999, p. 898. [14] P.A. Doyle, P.S. Turner, Acta Crystallogr. A 24 (1968) 390. [15] C.J. Humphreys, P.B. Hirsch, Philos. Mag. 18 (1968) 115. [16] J.M. Zuo, J.C.H. Spence, Ultramicroscopy 35 (1991) 185. [17] C. Deininger, G. Necker, J. Mayer, Ultramicroscopy 54 (1994) 15. [18] C. Birkeland, R. Holmestadt, K. Marthinsen, R. Hoier, Ultramicroscopy 66 (1996) 89. [19] R. Holmestadt, C. Birkeland, K. Marthinsen, R. Hoier, J.M. Zuo, Microsc. Res. Tech. 46 (1999) 130.
Crystal thickness and extinction distance determination using energy filtered CBED pattern intensity measurement and dynamical diffraction theory fitting D. Delillea,*, R. Pantelb, E. Van Cappellenc a
FEI France, 22 avenue Descartes, BP 45, F-94454 Limeil-Brevannes, France b ST Microelectronics, 850 avenue Jean Monnet, F-38926 Crolles, France c FEI Company, 7451 N.W. Evergreen Parkway, Hillsboro, OR 97124-5830, USA Received 27 March 2000; received in revised form 8 August 2000
Abstract A new method for measuring thickness and extinction distance of single crystals based on computed adjustment of measured and calculated CBED pattern intensity profiles is presented and discussed. The experimental beam intensity distribution is measured from an energy filtered CBED pattern recorded on a CCD camera. The calculated profile is based on dynamical diffraction theory, and with the two-beam approximation the analytical expression contains only two free parameters: specimen thickness t and extinction distance xg . Parameter refinement through minimization of the difference between experimental and calculated intensity profiles is carried out using OriginTM 5.0 software from Microcal. The iterative procedure always converges to a unique solution in a few seconds, yielding an accurate value for both thickness and extinction distance. The method is extensively tested on silicon using the (0 0 4) Bragg reflection. On specimens in the usual TEM thickness range, the method gives result similar to the conventional (P.M. Kelly et al., Phys. Stat. Sol. A31 (1975) 771; S.M. Allen, Philos. Mag. A 43 (1981) 325) graphical methods, both based on the measurement of fringe spacing. Moreover, it is shown that the calculation matches perfectly both the positions of the minimums and maximums as well as the amplitude of maximums. For any single intensity profile, specimen thickness and extinction distance can be determined with a precision of about 0.2%. A statistical comparison of our method with the Kelly and Allen techniques, based on more than 50 experiments, shows an improvement in measured extinction distance dispersion. Using 197 keV electrons, and liquid-nitrogen cryo-holder, the new technique yields an experimental value of 161 3 nm for the extinction distance for silicon with the (0 0 4) Bragg reflection. The equivalent tabulated value at 0 K is about 156 nm. Using the Kelly and Allen methods, the extinction distance is found to be 162 6 nm. The improvement in precision is a direct consequence of matching the intensity profile envelope, which contains information on the extinction distance. Also the accuracy of thickness determination is improved and is around 0.5 to 1% for common specimen thickness. The minimum measurable sample thickness is shown to be two to three times thinner than with the Kelly and Allen methods (0.3xg as opposed to 0.8xg ). With no independent calculation of the extinction distance needed, the method is also applicable on unknown crystals. The method is fast, simple and can be easily automated. # 2001 Elsevier Science B.V. All rights reserved. Keywords: CBED; Energy filtering; Extinction distance; Thickness measurement
*Corresponding author. Tel.: +33-4-76-76-45-84; fax: +33-4-76-76-45-99. E-mail address: [email protected] (D. Delille). 0304-3991/01/$ - see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 9 9 1 ( 0 0 ) 0 0 0 6 7 - X
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1. Introduction
Maximums : tanðpDktÞ ¼ pDkt;
Accurate specimen thickness measurements are needed for a wide variety of well established TEM applications ranging from image simulations (HRTEM) for validation of proposed structure models to correction schemes for absorption and fluorescence effects in quantitative EDX analysis and beam broadening calculations to determine the attainable analytical spatial resolution. Direct sample thickness measurements can be carried out using projected interface observations at different angles. Electron energy loss spectroscopy (EELS) can also yield specimen thickness [1], provided that the mean free path for inelastic scattering can be determined independently. Convergent beam electron diffraction (CBED) as a means to measure crystal thickness was used as early as 1940. MacGillavry [2] used two beam dynamical diffraction theory to link the fringe minimums observed in CBED patterns and first described by Kossel and Mo¨llensted [3], to the specimen thickness:
defining a set of Dkt ¼ xk0 :
ðs2i þ 1=x2g Þt2 ¼ n2i ;
ð1Þ
where si is the deviation of the ith minimum from the exact Bragg position, xg is the extinction distance and ni an integer number. Ackermann [4] showed that the specimen thickness t could be obtained from the slope of a plot of s2i versus n2i . Kelly et al. [5] first and Allen [6] later, described a more elaborate plotting technique for thickness determination. In the two beam approximation, curve for the diffracted beam intensity the rocking Fg 2 is given by [7]: 2 Fg ¼ sin2 b sin2 ðpDktÞ; ð2Þ b ¼ tan1 ð1=sxg Þ;
ð3Þ
Dk ¼ ½ð1 þ ðsxg Þ2 1=2 x1 g
ð4Þ
By differentiation with respect to s, the expression for the minimums and maximums of 2 Fg can be found Minimums : Dkt ¼ integer; defining a set of Dkt ¼ nk
ð5Þ
ð6Þ
Kelly et al. [5] expressed Eq. (5) as ðsi =nk Þ2 ¼ ð1=xg Þ2 ð1=nk Þ2 þ ð1=tÞ2 :
ð7Þ
Allen (1981) expressed Eq. (6) as ðsi =xk0 Þ2 ¼ ð1=xg Þ2 ð1=xk0 Þ2 þ ð1=tÞ2 :
ð8Þ 2
Kelly et al. [5] proposed a plot of ðsi =nk Þ versus ð1=nk Þ2 and Allen [6] a plot of ðsi =xk0 Þ2 versus ð1=xk0 Þ2 . Eq. (7) and (8) do plot the same straight line characterized by a slope equal to ð1=xg Þ2 and an intersection with the Y-axis (s ¼ 0) equal to ð1=tÞ2 . By measuring the slope and the intersection with the Y-axis, the extinction distance and the sample thickness are derived. However, both methods (Kelly and Allen) are subject to errors when wrong sets of nk or xk0 are chosen to match thickness and extinction distance. Some graphical solutions yield too large values for xg , and too small values for t, this happens when the curve is slightly (usually unnoticeably) curved with a negative second derivative (concave with respect to the X-axis). Inversely too small values for xg , and too large values for t will result when the second derivative is positive. The right solution is a straight line giving the correct value for the effective extinction distance and the specimen thickness. However, Allen has shown that multiple beam diffraction effects can induce distortions and that anomalous absorption displaces minimums and maximums differently. Particular experimental conditions can lead to a difficult choice for the right solution using the Kelly and Allen graphical plot, for example when a low number of minimums and maximums are used and when their positions are measured with low accuracy. By knowing the extinction distance the choice for the best set of nk or xk0 is made easier. The extinction distance can be calculated independently using the expression xg ¼ pVc cos yB =lFg ; where Vc is the volume of a unit cell, and Fg is the structure factor FðyB Þ for the g reflection. Calculating Fg necessitates the atomic scattering factors f ðyÞ, which are tabulated in the International
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Crystallographic Tables III [8]. When these data are not easily available, or if the observed material is unknown, the extinction distance cannot be calculated. Under such conditions, a doubt may subsist when choosing a plotting set for a Kelly or Allen graph. The inherent limitation of the Kelly and Allen methods is that the information carried by the intensity is not used. Only the fringe positions are measured and a limited number of points are considered, i.e. the maximums and minimums. The diffracted beam intensity exhibits oscillations convoluted a damping envelope. The with 2 expression for Fg calculated with the two-beam approximation dynamical diffraction theory (Eq. (2)), describes these oscillations. The ½1 þ ðsxg Þ2 1 term describes the envelope and the full-width at half-maximum of the envelope equals 2=xg . Obviously, intensity measurement will enhance and complement the information contained in the fringe spacing. The intensity distribution specifically contains valuable information about the extinction distance. In this communication we propose a method to determine both specimen thickness and extinction distance, using a quantitative measurement of diffracted beam intensities combined with theoretical modeling. The tools used to measure the CBED pattern intensity distributions are now commonly available on modern TEMs (i.e. energy filter and CCD camera). A CCD camera exhibits a linear dynamic range of about 104 counts and an energy filter is used to eliminate the inelastically scattered electrons so as to improve the CBED patterns. An intensity profile is extracted from the diffraction pattern, using a two-beam CBED condition. The transmitted beam and diffracted beam are recorded simultaneously with 2a single exposure. Then the intensity profile Fg as function of the deviation from exact Bragg diffraction is compared with Eq. (2). Only two free parameters t and xg are used to refine experimental and calculated profiles. The data analysis software OriginTM 5.0, from Microcal Inc., running on a personal computer, is used to fit the experimental CBED profile, with a non-linear least-squares fit method. This method is based on a modified Levenberg–
7
Marquardt algorithm [9], which is the most widely used among non-linear functions refinement methods. It allows to easily determine both parameters. The method is evaluated on a silicon crystal, using the (0 0 4) Bragg reflection and examples of typical results are presented in order to illustrate the real potential and advantages of this technique. A study is made as function of specimen thickness, down to 30% of the extinction distance, and an extensive comparison between the Kelly and Allen methods and ours is carried out using the silicon (0 0 4) Bragg reflection and involving more than 50 experiments with varying sample thickness. Particular attention was paid to extinction distance determination, as this parameter is supposed to be constant for all experiments, and also because it can be calculated independently. This comparison gives a clear evaluation of the precision of thickness and extinction distance measurements for all methods.
2. Experimental details The experiments are carried out on a Philips CM200 FEG microscope, equipped with a gatan imaging filter (GIF). To avoid radiation damages and possible carbon contamination when a small probe is focused on the specimen, a cooling holder is used at liquid-nitrogen temperature. The nanoprobe mode is used to produce a small probe with a 4.5 mrad convergence semiangle. A silicon specimen exhibiting a very shallow wedge angle and prepared using a focused ion beam (FIB) [10] is oriented into a two-beam diffraction condition, defined by the (0 0 4) Bragg reflection. Multiple CBED experiments are carried out perpendicular to the wedge edge, in order to explore a large range of sample thickness. The diffraction camera length is chosen so as to fit both the transmitted and the diffracted discs on the same image. Inelastically scattered electrons are eliminated using an energy slit of 10 eV centered around 197 keV elastic energy. Quantitative CBED patterns are acquired on the 1 K 1 K CCD camera of the GIF and an intensity profile perpendicular to the fringes is extracted from each CCD image, using Digital MicrographTM 2.5
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software. Pixel averaging, parallel to the fringes, is used to improve the signal-to-noise ratio. Finally, the profile inside the diffracted disc is plotted versus the deviation from the exact Bragg conditions, calibrated, with respect to the Bragg angle 2yB between the transmitted and the diffracted disc centers. Therefore, the experimental profile represents 2 Fg ðsÞ with s ¼ ðDy=2yB Þðl=d 2 Þ: hkl This experimental profile containing typically 300 measured intensities (i.e. 300 pixels) is compared with calculation using the analytical expression of Eq. (2). The best parameter solution (extinction distance xg and specimen thickness t) is determined using OriginTM software running under Windows NTTM and the iteration uses a non-linear Levenberg–Marquardt least-squares fit algorithm. All measured intensities have the same weight in the least-squares-iteration. Notwithstanding the fact that the intensity profiles are acquired using an energy filter, intensities corresponding to the minimums are never reduced to zero, and this is in contradiction with the dynamical theory of electron diffraction. This phenomenon is due to several effects: multiple scattering in the bulk crystal, plasmon scattering, thermal and surface effects, artifacts related to amorphous layers left by the FIB preparation technique and possible surface contamination. In the non-linear
least-squares refinement, this background is modeled by the sum of a constant base line and a broad symmetrical Gaussian centered on s ¼ 0. The least-squares iterative calculation needs a few seconds to find a unique solution for xg and t, and it also yields a relative accuracy figure. The complete process can be automated with only limited operator input after the CBED pattern acquisition. The described method besides being more accurate is also easier and faster than the Kelly and Allen plot methods.
3. Experimental results and data processing To obtain a (0 0 4)/(0 0 0) two-beam condition in silicon, the crystal is roughly tilted by 108 starting from the h1 1 0i zone-axis orientation. The convergent electron beam diffraction patterns are zero-loss filtered to eliminate the inelastically scattered electrons (see Fig. 1). The convergence semi-angle is adjusted to 4.5 mrad, so as to maximize the disc diameters without overlapping the discs and the camera length is chosen to use a maximum number of pixels of the 1 K 1 K CCD camera. The exposure time is adjusted in order to reach saturation of the CCD camera in the transmitted beam (around 10 000 counts). A line profile across the two disc centers is generated using Digital MicrographTM
Fig. 1. Convergent beam electron diffraction pattern of a 190 nm thick silicon crystal, acquired on a CCD camera, using 197 keV electrons. Bright diagonal lines appearing on the right-hand side of the (0 0 4) disc make part of the (0 0 2) disc intensity, overlapping (0 0 0) and (0 0 4) diffraction discs.
D. Delille et al. / Ultramicroscopy 87 (2001) 5–18
software. Transverse pixel averaging is used to improve statistics and increases the signal to noise ratio to about 103. Fig. 2 is the line profile intensity obtained from Fig. 1. This profile can be calibrated in inverse nanometers using the Bragg angle value between the central local maximum of the transmitted disc and the local minimum of the (0 0 4) disc. The deviation parameter s becomes 2 s ¼ ðDy=2yB Þðl=dhkl Þ:
Fig. 3 shows the intensity profile of (0 0 4) diffracted disc as a function of the deviation parameter. The width of the profile is about 300 pixels and a high signal-to-noise ratio is observed. These are the optimum conditions for an accurate comparison between calculated and measured intensities. Two-beam dynamical diffraction theory predicts zero intensity for the minimums, whereas the experimental curve shows 10–15% residual intensity under the minimums close to s ¼ 0. Energy filtering helps to eliminate inelastically scattered electrons that contribute to the experimental background. However, some residual background will remain because of anomalous absorption, non-perfect filtering (the slit is not infinitely small) and a camera point spread function that is far from the ideal delta function. Moreover, as described by Peng et al. [11] and simulated by Dudarev et al. [12], a part of this non-zero background is due to plasmon scattering.
Fig. 2. Profile obtained from Fig. 1, with an average of 50 pixels width across (0 0 0) and (0 0 4) diffraction discs.
9
From the maximums and minimums of Fig. 3, a Kelly and Allen plot can be generated (see Fig. 4). Because of the optimum experimental conditions, these measurements are a clear test of the performance and limitations of the Kelly and Allen graphical method. Fig. 4 presents a Kelly and Allen plot of ðsi =nk Þ2 versus ð1=nk Þ2 and ðsi =xk0 Þ2 versus ð1=xk0 Þ2 , for two sets of nk and xk0 series. From these two sets of possible solutions, the best one is the arguably straightest line which gives a specimen thickness equal to 198.5 nm and an extinction distance equal to 159.6 nm.
Fig. 3. Intensity profile of the (0 0 4) diffraction disc, extracted from Fig. 2, and normalized versus the deviation from the exact Bragg reflection.
Fig. 4. Graphical Kelly and Allen graphical method applied to the profile of Fig. 3.
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The theoretical value for the extinction distance of the 0 0 4 Bragg reflection in silicon for an electron beam energy of 197 keV can be calculated using the following expression: xg ¼ pVc cos yB =lFg ; where Vc ¼ a3 ¼ 160:2 A˚3, y ¼ 0:538; Fg ¼ 8f ðyB Þ; f ðyB Þ ¼ 1:605 A˚, and l¼ 2:51 102 A˚. This yields a theoretical value for extinction distance equal to xg ¼ 156 nm. The experimentally determined xg value of 159.6 nm is in good agreement. The difference of around 2% can be due to the Debye–Waller factor ðexpðB2g =4ÞÞ, although small at liquid-nitrogen temperature and other experimental conditions affecting the extinction distance such as weak multiple beam effects and plasmon excitations. The next figure shows the new technique applied on the experimental intensity profile of Fig. 3. OriginTM 5.0 software is used to compare the experimental profile with a calculated one using Eq. (2) and the difference is minimized using t and xg as adjustment parameters. To cope with the non-zero intensity of the experimental minimums, a background is first subtracted using a large smooth function. As described in Fig. 5, this smoothing function consists of a base line component and a unique Gaussian function. Sources of this background are of different nature. As mentioned above, silicon wedges are FIB prepared, and this technique is known to leave amorphous layers on both sides of the sample [13]. Diffraction of amorphous layers yields rings related to the local mean atomic distances. These rings are broad and have a low intensity overlapping the diffraction discs. This can contribute to the background measured in the diffracted disc, although no significant changes were observed in the background between samples of variable amorphous surface layers (obtained using different FIB acceleration voltages [13]). Another source of this background can be the convolution of the real sample transfer function (diffraction plus anomalous interactions) with an instrumental transfer function. The latter one contains the point-spread function of the CCD camera, which can be neglected while considering the pixel number used in each line scan. Moreover, the instrumental
transfer function has far less impact in diffraction mode compared to imaging mode. Main contribution to this background can be found in anomalous absorption, well described by Hirsch et al. [7] and modeled by Doyle and Turner [14], and Humphreys and Hirsch [15], and in plasmon scattering, first noted by Peng et al. [11]. Most of these plasmon interactions are filtered using the Gatan GIF, but, due to the finite size of the filter slit, plasmon contribution must be taken into account. Hirsch et al [7] described the case of an absorbing crystal using the following expression: 2 Fg ¼ sin2 bðsin2 ðpDktÞ þ sinh2 ðpDk0 tÞÞ
expð2pt=x00 Þ;
ð9Þ
where b ¼ tan1 ð1=sxg Þ;
Dk0 ¼ ð1 þ ðsxg Þ2 Þ1=2 x01 g ;
with x0g and x00 being anomalous absorption distances. The term expð2pt=x00 Þ describes the intensity attenuation caused by sample thickness and is independent of s. The second term sin2 b sinh2 ðpDk0 tÞ is symmetric around s ¼ 0, has a maximum at s ¼ 0 and tends to zero for large s values. This shape roughly corresponds with the background observed in all the experiments. However, as the hyperbolic sine is not very different from a Gaussian function, we have chosen, for the sake of simplicity, a Gaussian background model. Fig. 5 shows good agreement between the experimental and calculated intensities. A solution yielding a similar extinction distance and specimen thickness than those obtained from the Kelly and Allen graph (Fig. 4) not only reproduces the right positions of the minimums and maximums, but also fits perfectly the oscillation amplitudes. This confirms that the two-beam dynamical theory perfectly applies to the experimental conditions used to obtain the profile of Fig. 3, and that the background subtraction does not introduce too many artifacts. The other advantage of a full statistical analysis is the accuracy estimate. In this case, xg ¼ 160:06 0:36 nm and t ¼ 198:55 0:24 nm, which means that we have a relative precision of about 2 103
D. Delille et al. / Ultramicroscopy 87 (2001) 5–18
11
Fig. 5. Two-beam dynamical diffraction calculated profile, and comparison with experimental CBED profile of Fig. 3. Gaussian and baseline extraction of the background are presented.
Fig. 6. Comparison between calculated and experimental CBED profiles, using fixed values of respectively 157 and 163 nm for xg (systematic 2% error around 160 nm), and 198.55 nm for t.
for the extinction distance and 103 for the specimen thickness. The envelope of oscillations in a two-beam condition is related to the extinction distance, and the width at half maximum of the envelope is equal to 2=xg . Therefore, the solution for xg should be accurate and unique, in order to reproduce the right sequence of decreasing amplitude. This hypothesis can be tested by arbitrarily fixing the extinction distance within 2% of its best-estimated value and check whether a match is possible between experiment and theory. Fig. 6 shows fits using xg ¼ 157 nm and xg ¼ 163 nm, both with the same optimum sample thickness of 198.5 nm. This clearly shows that a non-optimized xg cannot explain the right amplitudes of the maximums, and that the fitting process is sensitive to 0.2% variation of the extinction distance value. If both xg and t are varied, neither the amplitude of maximums nor the positions of maximums and minimums are accurately fitted. In this way, Zuo and Spence [16] noticed that under two-beam diffraction conditions, the intensity at s ¼ 0 was strongly dependent on xg variations, with a maximum sensitivity for an optimum thickness of t ¼ 1:25xg . In this range, a 1% variation in xg gives a 6% variation in intensity. Fig. 6 shows very similar results, with an intensity variation of about 11% for a 2% xg variation. This confirms a high relative precision
of 0.2% with which both xg and t are measured in Fig. 5. Figs. 7a–e are examples of CBED patterns acquired at different thickness on a silicon wedge, ranging from 2xg down to 0:3xg , and Figs. 8a–e are the corresponding experimental and calculated intensity profile fits together with the Kelly and Allen plots. Table 1 summarizes the comparison between graphical method and intensity fit method results. For specimen thickness range above 0:8xg the two methods give similar results with less than 2% difference. Nevertheless, Kelly and Allen methods give only a set of possible solutions, instead of the unique solution obtained using the intensity fit method. Below 0:8xg , the extinction distance determined by graphical Kelly and Allen plot becomes unstable (see dispersion in Fig. 8b(2)). Under 0:6xg , no acceptable solution is found by this way for xg and therefore no solution for thickness t. By comparison, an accuracy of 1% is still obtained on thickness determination below 0:3xg using the intensity fit method. Fig. 9a is a graph of the extinction distance as function of sample thickness using the Kelly and Allen plot method. The measured extinction distances are dispersed around a mean value, which tends to increase with decreasing thickness. Above a sample thickness of 130 nm, which corresponds to 0:8xg , the mean value is 162 10 nm and 70% of the data points are
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Fig. 7. (a)–(e). Series of 197 kV experimental CBED patterns, showing the (0 0 4) diffracted and (0 0 0) transmitted discs for different silicon crystal thickness.
D. Delille et al. / Ultramicroscopy 87 (2001) 5–18
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Fig. 8. a(1) Intensity profile fit corresponding to Fig. 7a: extinction distance xg ¼ 163:65 nm and thickness t ¼ 280:97 nm. a(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 281 nm thick silicon crystal. b(1) Intensity profile fit corresponding to Fig 7b: extinction distance xg ¼ 160:14 nm and thickness t ¼ 150:17 nm. b(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 150 nm thick silicon crystal. c(1) Intensity profile fit corresponding to Fig 7c: extinction distance xg ¼ 166:00 nm and thickness t ¼ 125:47 nm. c(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 125 nm thick silicon crystal. d(1) Intensity profile fit corresponding to Fig 7d: extinction distance xg ¼ 164:85 nm and thickness t ¼ 94:21 nm. d(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 95 nm thick silicon crystal. e(1) Intensity profile fit corresponding to Fig 7e: extinction distance xg ¼ 160:28 nm and thickness t ¼ 45:30 nm. e(2) Graphical refinement of xg and t, using Kelly and Allen method, applied to a 45 nm thick silicon crystal.
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Fig. 8. Continued.
Table 1 Comparison between Kelly and Allen methods and ours, based on results of Fig. 8. Thickness range 2 xg xg 0:8 xg 0:6 xg 0:3 xg
This work
Kelly and Allen
Comments
t (nm)
xg (nm)
t (nm)
280.97 0.29 150.17 0.15 125.47 0.06 94.21 0.40 45.30 0.43
163.65 0.27 160.14 0.35 166.00 0.14 164.85 4.46 160 48
280 163 152 163 125 166 94 185 No coherent solution
concentrated around 162 6 nm. Under 130 nm thickness, the extinction distance starts to increase, and below 110 nm or 0:7xg the measured values become unacceptable. To conclude, the Kelly and Allen plot methods allow measuring the extinction
xg (nm) Very similar results for the two methods. Some dispersion in Kelly and Allen plot. xg starts to diverge using Kelly and Allen method. Thickness determination with 1% accuracy using intensity fit method.
distance with a 4% relative precision for samples thicker than 0:8xg . Fig. 9b handles the same data as Fig. 9a, but using the intensity fit method. Below 100 nm thickness the error bars are indicated. An increase
D. Delille et al. / Ultramicroscopy 87 (2001) 5–18
15
Fig. 9. (a) Plot of extinction distance versus crystal thickness : each point represents one of the 50 silicon crystals analyzed using Kelly and Allen graphical method. (b) Plot of extinction distance versus crystal thickness : each point represents one of the 50 silicon crystals analyzed using our non-linear least-squares fit method. (c) Histograms extracted from graphs (a) and (b). The two dashed curves are the fitted Gaussian functions corresponding to each histogram. The xg step (i.e. bar width) is 1 nm.
in the extinction distance at low thickness observed while using the Kelly and Allen plot method is virtually non-existent. Above 130 nm, the mean value is 161 5 nm and most of the values (70%) are concentrated around 161 3 nm. Above 130 nm thickness or 0:8xg , the intensity fit method allows to measure the extinction distance
with a 2% relative precision. For 100 and 50 nm thickness the error drops to 3 and 6% respectively, without significantly changing the value. Not only does accuracy and precision of extinction distance and sample thickness measurements improve, when using the intensity fit method, but also the minimum usable sample thickness is decreased
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from 0:8xg to 0:3xg when compared to the Kelly and Allen technique. The histograms of Fig. 9c represent the extinction distances obtained by using both methods and only the most favorable sample thickness range between 130 and 230 nm was considered. This leaves a total of 36 measurements. The intensity fit method clearly yields a narrower distribution than the Kelly and Allen plot method. Statistical analysis of the histograms through Gaussian distribution fits confirms the better accuracy of the intensity fit technique and gives the following numbers: xg ¼ 161 3 nm for the new method and xg ¼ 162 6 nm for the conventional method. To conclude, the accuracy in extinction distance measurement is around 2% using the new intensity fit technique, provided that the sample thickness is above 0:8xg . With the Kelly and Allen method it is generally accepted that the error on the thickness measurements is less than on the extinction distance measurements. The same holds for the intensity fit technique and as such it is fair to say that the precision in thickness measurement is around 0.5– 1%.
4. General discussion In our simulation procedure, we only have considered the true two-beam conditions to simulate the theoretical intensity profiles. Considering multiple beam effects necessitates the use of the matrix formulation of Bloch waves intensity [7,16–19], and the anomalous absorption modeling involves the use of complex atomic potentials, leading to the use of a large number of parameters. The aim of our work is to propose a thickness measurement method as simple as the Kelly and Allen ones, but more accurate and easy to use. As shown by Gjonnes et al. [19], we replace the Uð0 0 4Þ coefficient by its effective value, in order to fit experimental profiles. So, the extinction distance and structure factor values refined using our method are effective values, shown by Gjonnes et al. [19] to be very near absolute values using the two-beam approximation. So we suggest to choose the best two-beam orientations in order to limit
the refinement conditions around two key parameters: t and xg . In the case of the (0 0 4) reflection in a diamondlike crystal structure, the two-beam conditions are easily satisfied because the systematic Bragg reflections (0 0 2), (0 0 2) and (0 0 6) are not present on the diffraction pattern (structure factor equal to zero). The only possible reflections, (0 0 4) and (0 0 8) have a very high deviation parameter s, and their intensities are negligible. Experimental care are, essentially, sample cooling at liquid nitrogen temperature, reduced beam exposure time and minimization of the related carbonaceous contamination, and of course energy filtering. In such a condition, the background is reduced to its minimum value (10–15% of the total intensity), but is still present in the intensity profiles. In order to simplify our parameters refinement, we have modeled this background as a gaussian one instead of considering the anomalous absorption and plasmon scattering [11,12] and their heavy formalism. According to the experimental results presented above, our intensity fit method yields similar results as the graphical methods of Kelly and Allen, albeit with a better accuracy and precision and it is applicable to thinner samples. Kelly and Allen have estimated the accuracy of their thickness measurements to be around 2%, at least for a sample of reasonable thickness. They consider three minimums or maximums in the diffraction disc to be the lower limit of their technique. In this case they suggest that the extinction distance be measured in a thicker part of the specimen and to use this value in the plot. While using our intensity fit technique, the OriginTM 5.0 software matching results for a particular experiment shows a relative accuracy of about 0.1–0.2% for both the specimen thickness and extinction distance on the condition that the sample thickness is in the range between 0:8xg and a few xg . However, when a large number of experiments are carried out (Fig. 9b), the extinction distance accuracy drops to around 2%. The origins of this dispersion are the cumulative effects of experimental errors. The major contributions are accuracy of xB measurement (0.2% 1 pixel per
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500 pixels), image distortion in the energy filter and CCD camera (less than 0.3%), errors due to background modeling (probably less than 0.5%) and variable beam-induced thermal effects (around 1%). The improvement in extinction distance accuracy using our intensity fit method, over the Kelly and Allen method, is a factor of two for specimen thickness in the range of 0:8 xg to few xg (2% instead of 4%). For thinner samples, the difference becomes even more significant as the intensity fit method can measure xg on samples down to 0:3 xg , whereas the Kelly and Allen technique becomes virtually useless below a thickness of 0.8 xg . A statistical analysis determined the extinction distance of the (0 0 4) diffracted beam in silicon at 197 keV to be 161 nm 2% (Fig. 9c), whereas the value calculated from tabulated atomic scattering factors equals 156 nm. The difference of 3% might be explained by thermal agitation described by the Debye–Waller factor expðB2g =4Þ, although small at liquid nitrogen temperature. Anomalous absorption and multiple beam effects also influence the effective extinction distance. The same extinction distance measured using a Kelly and Allen graph yields 162 nm and the standard deviation is 4%. The accuracy with which sample thickness can be measured is around 0.5 to 1% for intensity fit technique even for very thin samples (0:3xg ), whereas the Kelly and Allen method fails for specimen is of thickness below 0:6xg . The Kelly and Allen plot techniques limit the information extracted from the diffracted disc to the fringe positions (the positions of minimums and maximums). The intensity information is lost. The two-beam dynamical diffraction approximation allows, besides predicting the positions of the minimums and maximums, to use the intensity information. The envelope of intensity profile oscillations is directly related to the extinction distance since the full-width at halfmaximum equals 2=xg . Beyond the envelope, the rocking curve and the extinction distance are intricately connected and as such modeling the measured intensities yields accurate information about xg . Moreover, Zuo and Spence [16] have noticed that the intensity at exact Bragg angle
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(i.e. s ¼ 0 in the intensity profile) is strongly xg sensitive. Fig. 6 shows the extreme sensitivity of the rocking curve to the extinction distance and for a particular intensity profile, a unique value with an accuracy of 0.2% can be found. This is a distinct advantage over the Kelly and Allen plot methods where wrong choices for the extinction distance can be made, which is particularly problematic when the extinction distance cannot be calculated from tabulated atomic scattering factors. This is especially true when dealing with unknown crystals and a wrong extinction distance will lead to a significant error in the thickness measurements. With the intensity fit method it is straightforward to measure extinction distance and thickness of an unknown crystal, as only the Bragg angle must be determined.
5. Conclusions A new procedure for mono-crystal specimen thickness and extinction distance measurements using convergent beam electron diffraction is presented. Energy filtered and digitally recorded CBED patterns are used to produce the experimental intensity profile (the so-called rocking curve) which is fitted using the non-linear leastsquares refinement of a simulated one. The calculated profile consists of a two-beam dynamical approximation rocking curve and a background to cope with the experimental conditions. The iterative procedure optimizes two fit parameters, the extinction distance and the specimen thickness, and always converges towards a unique solution in a few seconds. The match between experimental and predicted is extraordinary and the accuracy of thickness measurements is between 0.5 and 1%. The method also allows to process very thin samples not suited for the conventional techniques. The effective extinction distance is also measured with a high accuracy (2%). Recent developments such as CCD cameras and processing software have made the method both fast and simple, but further automation can still be envisaged. The method is also applicable on
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unknown crystals since no independent calculation of the extinction distance is needed.
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