Microscopy Microanalysis
Microsc. Microanal. 20, 111–123, 2014 doi:10.1017/S1431927613013913
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© MICROSCOPY SOCIETY OF AMERICA 2014
Sample Thickness Determination by Scanning Transmission Electron Microscopy at Low Electron Energies Tobias Volkenandt,* Erich Müller, and Dagmar Gerthsen Laboratorium für Elektronenmikroskopie, Karlsruher Institut für Technologie (KIT), Engesserstr. 7, 76131 Karlsruhe, Germany
Abstract: Sample thickness is a decisive parameter for any quantification of image information and composition in transmission electron microscopy. In this context, we present a method to determine the local sample thickness by scanning transmission electron microscopy at primary energies below 30 keV. The image intensity is measured with respect to the intensity of the incident electron beam and can be directly compared with Monte Carlo simulations. Screened Rutherford and Mott scattering cross-sections are evaluated with respect to fitting experimental data with simulated image intensities as a function of the atomic number of the sample material and primary electron energy. The presented method is tested for sample materials covering a wide range of atomic numbers Z, that is, fluorenyl hexa-peri-hexabenzocoronene ~Z ⫽ 3.5!, carbon ~Z ⫽ 6!, silicon ~Z ⫽ 14!, gallium nitride ~Z ⫽ 19!, and tungsten ~Z ⫽ 74!. Investigations were conducted for two primary energies ~15 and 30 keV! and a sample thickness range between 50 and 400 nm. Key words: scanning transmission electron microscopy, sample thickness determination, Monte Carlo simulation, scattering cross-section, quantification, low energy
I NTR ODUCTION Knowledge of the local sample thickness is essential in transmission electron microscopy ~TEM! for the quantification of any image information. For this reason, several methods have been developed to determine the local TEM sample thickness. Plasmon losses obtained from electron energy loss spectra ~Egerton, 1986!, convergent beam electron diffraction ~Williams & Carter, 2009!, electron holography ~Lehmann & Lichte, 2002!, and thickness contours under well-defined excitation conditions ~Williams & Carter, 2009! can be exploited as well as scanning transmission electron microscopy ~STEM! at TEM-typical energies ~Rosenauer et al., 2009; Van den Broek et al., 2012!. These techniques vary in precision, complexity, and spatial resolution. More recently, STEM in the high-angle annular darkfield ~HAADF! mode at primary energies E0 ⱕ 30 keV was suggested as a promising technique to determine the TEM sample thickness ~Morandi & Merli, 2007; Krzyzanek & Reichelt, 2008; Volkenandt et al., 2010; Pfaff et al., 2011; Klein et al., 2012!. This method is based on the comparison of experimental and calculated HAADF STEM intensities. It can be applied if the composition of the sample material is known. It is advantageous that low-energy STEM images can be recorded quickly in an easy-to-operate SEM fitted with a STEM detector. Combining low-energy STEM in an SEM with a focused ion beam ~FIB! system for TEM sample preparation facilitates thickness quantification directly after FIB sample preparation. Moreover, knock-on damage is negligible because of the low electron energy in an SEM. Spatial resolution is in the range of 1 nm because of the small Received August 1, 2013; accepted November 6, 2013 *Corresponding author. E-mail: [email protected]
interaction volume for thin samples, but decreases with increasing sample thickness because of beam broadening. The quantification of low-energy HAADF STEM data requires an adequate description of the experimental HAADF STEM intensities. Monte Carlo ~MC! simulations are well established for the calculation of the low-energy HAADF STEM intensity because effects of Bragg diffraction can be neglected at these low electron energies and large scattering angles ~Treacy & Gibson, 1993!. However, the results of MC simulations depend on the scattering cross-sections used. Depending on the atomic number of the sample material and the electron energy screened Rutherford cross-sections ~SR-CS! ~Rutherford, 1911; Bishop, 1967! or Mott crosssections ~M-CSs! ~Mott & Massey, 1949! can be used. Different methods were applied to estimate the effects of atomic structure and screening potentials. This leads to a variety of scattering equations, which cannot be solved analytically. Depending on the approximations the equations are only valid for a limited range of parameters ~Motz et al., 1964!. In general, SR-CSs adequately describe electron scattering for low atomic number ~low Z! materials while M-CSs are used for higher atomic numbers. Browning et al. ~1994! stated that SR-CSs give acceptable results for high electron energies and low-Z materials. M-CSs are applied for low to intermediate incident energies ~100 eV–30 keV! and high-Z materials. An estimate on the accuracy is not straightforward. Only comparison with experimental data can confirm the validity of a particular model for the calculation of scattering cross-sections. In this work we focus on the application of low-energy HAADF STEM for TEM sample thickness determination. The measured HAADF STEM intensities are normalized with respect to the intensity of the incident electron beam
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and can thus be directly compared with simulated intensities. For this purpose, wedge-shaped samples with a wellknown thickness profile were prepared by FIB milling. The method is tested for different materials covering a wide range of ~average! atomic numbers between 3.5 @fluorenyl hexa-peri-hexabenzocoronene ~FHBC!# and 74 ~tungsten!. The experimental results are compared with MC simulations on the basis of screened SR-CSs and M-CSs, which yield information on the applicability of the scattering cross-sections as a function of atomic number at electron energies between 15 and 30 keV.
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Experimental Setup Sample preparation and STEM imaging were performed with an FEI Strata 400S dual-beam microscope, which combines an SEM equipped with a Schottky field-emission gun and a focused Ga⫹-ion beam ~FIB! system. It is equipped with an FEI STEM III semiconductor detector positioned below the sample with a fixed distance of 18.8 mm between the objective lens pole piece and detector. The detector is divided into six ring-like segments where the outermost ring is the HAADF segment. The scattering angle range is determined by the inner and outer radii of the selected segment and the working distance between the sample and the objective lens pole piece. The latter was chosen as 6.5 mm for this work. Accordingly, electrons transmitted into a hollow cone between 187 and 683 mrad are detected by the HAADF segment. Primary energies of 15, 20, 25, and 30 keV were used for imaging to detect possible energy-related effects. Only the HAADF ring was used in this work. However, the presented approach can be also applied for the bright-field or annular dark-field ~ADF! segments covering smaller scattering angle ranges if conditions leading to Bragg diffraction are avoided ~e.g., by tilting!. In our case, the signals of the ADF segments were too low for the studied sample thickness range because of the small detection area of these segments. During imaging special attention has to be paid to the contrast ~gain! and brightness ~offset! settings, which are usually tuned to achieve optimum image contrast. While this is useful for enhancing the image contrast it is unfavorable for this work because the image intensity must be quantitatively comparable with MC simulations. To ensure comparability with the simulations, the intensity is normalized with respect to the incident beam. For normalization of the measured intensities we use the following procedure. The detector is first directly illuminated without sample to measure the intensity of the incident electron beam. Contrast and brightness are then adjusted such that the inactive bright-field and ADF detector segments appear black, that is, they are assigned with a gray level IB close to 0. In contrast, the active HAADF segment appears white, that is, it is assigned with a gray level IW close to 65,535 for our 16-bit detector. Oversaturation and undersaturation have to be carefully avoided during this procedure. Figure 1 shows such a detector reference
Figure 1. High-angle annular dark-field scanning transmission electron microscopy ~HAADF-STEM! image of the STEM detector. The active HAADF segments show up bright while all other segments appear dark. The marked areas illustrate the regions that were used to measure the black and white levels of the detector.
image, which was taken at the lowest possible magnification. Only a small, but for the contrast adjustment sufficient innermost part of the HAADF segment can be seen. Additionally, the sample holder ~which was retracted for this image! usually obstructs the upper half of the reference image. That is why the black and white levels are measured in the regions in the lower half of the image as marked in Figure 1. The detector settings remain unchanged for the subsequent images of the samples. The samples typically appear at intermediate gray levels, of about 30,000, depending on the sample thickness and primary electron energy. The intensity values of the sample IS are measured and subtracted by the black level IB measured before in the detector reference image. This offset correction is also performed for the measured white level IW . The normalized HAADF STEM intensity of the sample is denoted by IHAADF and given by the following equation: IHAADF ⫽
IS ⫺ IB IW ⫺ IB
.
~1!
IHAADF can be directly compared with MC simulations. Contrast tuning of the detector with respect to IB and IW is crucial for the intensity measurements in the following because it effectively allows an absolute measurement of the intensity. But this only holds if the amplification characteristic of the detector is linear. Therefore, this was checked by imaging the detector using a large range of beam currents and all primary electron energies of interest. Brightness and contrast were adjusted for the highest beam current and then kept constant. The image intensity of the selected detector segment was investigated as a function of the beam current, which was measured by a Faraday cup. A linear behavior was found between 10 pA and 10 nA, which confirms the linear response of the detector.
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To achieve agreement between measured and simulated intensities, the simulated HAADF intensities were corrected by taking into account the following detector-related effects. According to Reimer ~1998!, the signal of a semiconductor STEM detector in terms of the collection current ICC can be described by the following equation: ICC ⫽ N~1 ⫺ h!
E ⫺ Eoff Ehp
«,
~2!
with the number of electrons N impinging on the detector, the backscatter coefficient of the detector material h, the charge collection efficiency «, the average energy of the scattered electrons E, and the energy offset Eoff , which takes into account the energy loss of electrons while entering the detection layer. The energy for generating an electron-hole pair within the detection layer is denoted by Ehp . For the silicon-based STEM detector in this work we used « ⫽ 1 ~Reimer, 1998!, Ehp ⫽ 3.6 eV ~Scholze et al., 2000! and Eoff ⫽ 3 keV ~determined experimentally!, as confirmed in a previous study ~Volkenandt et al., 2010!. The backscatter coefficient h was calculated according to the following equation ~Reimer, 1998!: h ⫽ ~1 ⫹ cos u!⫺9 / M Z,
A0 ⫺ Ab A0
,
~4!
with the area of the complete HAADF ring A0 and the insensitive detector area Ab . The area of the complete HAADF ring is given by the inner and outer radii rmax and rmin of the detector in the following equation: 2 2 A 0 ⫽ p~rmax ⫺ rmin !,
~5!
Ab is determined by nb ⫽ 6 blind gaps with a width bb ⫽ 0.078 mm in the following equation: A b ⫽ nb bb ~rmax ⫺ rmin !.
IHAADF, s ⫽
~3!
with the scattering angle u and the atomic number Z ⫽ 14 for the detector material. In addition, a geometric effect has to be considered. The ring-like HAADF segment is divided into six subsections ~see Fig. 1!. Owing to this segmentation, there are insensitive regions on the detector, which do not contribute to the HAADF STEM signal. The real detection area is described by a geometric correction factor given in the following equation: cA ⫽
Figure 2. Simulated scattering angle distribution for C ~dashed line! and Pt ~solid line! films of 150 nm thickness at 15 keV primary energy. The gray vertical lines mark the angular range covered by the high-angle annular dark-field segment of the scanning transmission electron microscopy detector.
~6!
The corrections described by equations ~2–6! are applied during the integration of the scattering angle histograms obtained from the MC simulations. This yields the corrected simulated intensity on the HAADF segment Jcorr , which has to be normalized with respect to the incident intensity J0 for comparison with measured intensities. J0 is given by the number of incident electrons 10 6 corrected for the physical detector effects. Energy losses are not considered for J0 so that E equals E0 in equation ~2!. The simulated HAADF intensity IHAADF,s is then given by
Jcorr J0
,
~7!
which can be directly compared with the measured HAADF intensity IHAADF if the image was recorded under the abovementioned conditions for absolute intensity measurements.
MC Simulations MC simulations are well established for the calculation of SEM and STEM intensities at electron energies E0 ⱕ 30 keV. Multiple scattering prevails even in thin samples resulting from mean free path lengths in the order of 10 nm and below. MC simulations adequately describe the stochastic nature of the scattering process and are therefore widely used to predict low-energy STEM intensities. In this work we used the NISTMonte package ~Ritchie, 2005! with implemented SR-CSs ~Heinrich, 1981! and M-CSs ~Powell et al., 2005!. The energy loss of the electrons is calculated by the Joy–Luo formalism ~Joy & Luo, 1989!, which is a modification of Bethe’s continuous slowing down approximation. The samples in the simulations are modeled as an infinite, homogeneous film, with constant thickness. Simulation series for different film thicknesses and primary electron energies were performed. The number of incident electrons was chosen to be 10 6 to minimize the statistical error. The simulations yield the scattering angle and energy histogram of the scattered electrons. A typical scattering angle distribution is shown in Figure 2 for carbon and platinum with 150 nm sample thickness for 15 keV. The scattering angle range covered by the STEM detector is marked by vertical lines. As carbon is a weakly scattering material the angular distribution in Figure 2 shows a rather pronounced maximum at small scattering angles around 100 mrad. Platinum on the other hand scatters strongly because of its large atomic number. The angular distribution shows a small but
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broad maximum around 700 mrad. The more pronounced maximum at 2.4 rad stems from backscattered electrons whose number already exceeds the number of electrons, which are transmitted through the sample at 150 nm thickness. This is also recognized in the HAADF signal, which corresponds to the area under the curves between the vertical lines. IHAADF for carbon is larger resulting in brighter image contrast. An increase in thickness would cause the carbon curve to broaden and shift to higher scattering angles reaching a shape comparable to the actual platinum curve at large sample thicknesses. A decrease in thickness would cause the curves to sharpen and shift to lower scattering angles.
Sample Materials The samples were chosen to cover a wide range of atomic numbers Z to show the broad applicability of the method. The sample material with the lowest Z is FHBC, a polymer with the chemical formula C100H98 , an average atomic number of Z ⫽ 3.5, and density r ⫽ 1.1 g/cm 3. Five films with different thicknesses were prepared by spin coating. To determine the film thicknesses with an alternative method, a Dektak profilometer was used. These thickness values will be denoted as nominal thickness values. The precision of the Dektak thickness measurements is 615 nm, given by the standard deviation of the mean value after multiple measurements. Carbon ~Z ⫽ 6, r ⫽ 1.9 g/cm 3 ! was chosen as another low-Z sample material, which was deposited by electronbeam vapor deposition on a mica substrate as an amorphous layer. Using FIB-milling a cross-section lamella was prepared. Electron beam and ion beam induced Pt deposition was used for sample protection. The lamella was thinned to a thickness of about 1 mm, cut free, and transferred to a dedicated copper sample grid using an Omniprobe manipulator needle. The lamella was fixed by ion beam-induced Pt deposition. The lamella was further thinned in a wedgelike shape with a nominal wedge angle of 208 to obtain a well-defined thickness profile. The same FIB-based preparation technique was used to prepare wedge-shaped lamellae of Si ~Z ⫽ 14, r ⫽ 2.3 g/ cm 3 ! and GaN ~average atomic number Z ⫽ 19, r ⫽ 6.1 g/cm 3 !, which can both be considered as representatives of materials with intermediate atomic numbers. The silicon sample is a Si/Ge heterostructure, which was grown by reduced pressure chemical vapor deposition on a Si substrate. It contains four layers with varying Ge concentration embedded in the Si matrix. We did not study the Si/Ge layers because their Ge concentration was not known precisely enough. Only the Si matrix was analyzed in this work. The GaN sample was prepared from a commercial GaN wafer grown by hybrid vapor phase epitaxy. FIB preparation was performed using a nominal wedge angle of 308 for the Si and GaN samples. Tungsten ~Z ⫽ 74, r ⫽ 19.3 g/cm 3 ! was chosen as a representative of heavy materials. The preparation was again performed by FIB with a wedge angle of 208. All samples
Figure 3. 15 keV high-angle annular dark-field scanning transmission electron microscopy image of a fluorenyl hexa-perihexabenzocoronene film with a nominal thickness of 65 nm. White squares mark positions for intensity measurements at different film thicknesses.
were imaged at electron energies of 15, 20, 25, and 30 keV to check energy-dependent effects on the image intensity but only the results obtained at 15 and 30 keV are presented here.
E XPERIMENTAL R ESULTS Figure 3 shows a representative HAADF STEM image of one FHBC film ~Nr. 1! taken at 15 keV. The film does not lie flat on the Cu grid but turned out to be multiply folded. Regions with the lowest intensity correspond to a singlelayer film ~marked by 1 in Fig. 3! while folded regions can be recognized by their higher intensity ~2–5 indicate multiplelayer film regions in Fig. 3!. The folded film regions yield HAADF intensities at integer values of the nominal film thickness of film Nr. 1 of 65 nm. The thickness homogeneity of the FHBC film can be assessed by the homogeneous HAADF STEM intensity. Further, IHAADF values could be obtained from four more FHBC films ~Nr. 2–5! with nominal thicknesses of 90, 106, 121, and 244 nm. Figure 4 depicts IHAADF of all five FHBC films as a function of the nominal thickness for E0 ⫽ 15 keV ~Fig. 4a! and E0 ⫽ 30 keV ~Fig. 4b!. Solid symbols mark single-film IHAADF data ~for all five films!, while empty symbols indicate measurements at different positions of multiply folded film regions ~for films Nr. 1–3!. The dash-dotted line gives IHAADF,s on the basis of M-CSs, while the dashed line shows simulation results using SR-CSs. The experimental results for 15 keV ~Fig. 4a! can be well described by MC simulations based on SR-CSs. This does not seem to apply for the data points of film Nr. 2 ~diamond symbols in Fig. 4!, which, on first sight, can be better fitted with IHAADF,s based on M-CSs. This disagreement can be explained by the error for the nominal film thickness of film Nr. 2 of 90 nm and the error of the profilometer measurements of at least 615 nm. Assuming a smaller nominal thickness of 70 nm
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Figure 4. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities of five fluorenyl hexa-peri-hexabenzocoronene films as a function of the nominal film thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. Solid symbols mark measurements of single-layer film regions, empty symbols mark measurements at positions with multiple-layer film thickness because of folding of the film. The arrows in ~a! mark thickness-corrected IHAADF for film Nr. 2 as described in the text.
results in good agreement with the SR-CS-based simulations as indicated by arrows in Figure 4a. The results on this film demonstrate that the precision of HAADF STEM based measurements is better than the profilometer data. Figure 4b shows data for 30 keV with low intensities and a shallow slope of the IHAADF versus thickness curve. The experimental data can be well fitted by SR-CS-based simulations but discrepancies are observed for larger sample thicknesses. Clearly, lower electron energies are preferred for low-Z materials with respect to the accuracy of thickness determination. Figure 5a shows a 15 keV HAADF STEM cross-section image of the C wedge sample. The mica substrate, the C layer and the Pt protection layer can be well distinguished because of their different intensities. The sample thickness increases from the left to the right because of the wedge geometry along the white arrow, which correlates with the increase of IHAADF . At the thin edge a higher intensity of Pt is observed, as expected, but a contrast inversion occurs for Pt with increasing sample thickness. This effect can be explained by considering Figure 2 because fewer electrons are scattered in the HAADF angular range because of the large fraction of backscattered electrons. It is noted that the Pt protection layer consists of Pt and a considerable fraction of C, which prevents the quantification of the sample thickness in this case. A top-view secondary electron ~SE! SEM image of the same wedge is presented in Figure 5b, which reveals a wedge angle of a ⫽ 18 6 18 and a thickness offset at the thin edge of the wedge t0 ⫽ 50 6 15 nm. Only the capping Pt protection layer is visible in top-view. The sample material below may exhibit different sputtering rates during the FIB treatment, which in effect leads to uncertainties of a and t0 determined from SE SEM images. The white arrow in Figure 5a marks an intensity linescan averaged over a width of 20 px along the wedge starting at the edge. Owing to the wedge geometry each point x along the linescan can be
correlated with a local sample thickness t according to the following equation: t ⫽ t0 ⫹ x{tan a.
~8!
This procedure is performed for all wedge samples in the following. Simulated and experimental IHAADF values for carbon are shown in Figures 6a and 6b for 15 and 30 keV, respectively, which include simulations on the basis of M-CSs and SR-CSs. The experimental data ~black solid line! were derived from the intensity linescan with the position coordinate transformed into local sample thickness according to equation ~8!. The two parameters t0 and a were adjusted leading to horizontal shift and stretching of the linescan to achieve an optimum fit with one of the simulated curves. The intensity values of the linescan remain unchanged. The best fit was obtained for MC simulations based on SR-CSs for t0 ⫽ 70 nm and a ⫽ 17.28 for both electron energies. However, some discrepancies remain in particular for E0 ⫽ 15 keV. The measured intensities are too low, although the shape reproduces the SR-CS-simulated curve. This effect will be further considered in the Discussion section. A deviation at thicknesses below 80 nm is visible for both energies. It is caused by a blunting of the wedge edge, which can be seen in Figure 5b. Blunting of the edge effectively increases the wedge angle and leads to a steeper reduction of the sample thickness and IHAADF . Figure 7 presents a 30 keV HAADF STEM cross-section image of the Si wedge sample in Figure 7a and a top-view SE SEM image in Figure 7b. The four Si/Ge layers in Figure 7a show brighter contrast than the Si matrix. The layer directly under the Pt protection layer shows brighter contrast because of Ga implantation during the FIB preparation process. The sample thickness increases from left to right and IHAADF was measured in the Si matrix area as indicated by the white arrow.
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Figure 5. a: 15 keV high-angle annular dark-field scanning transmission electron microscopy image of the C wedge sample. The position of the intensity linescan is indicated by a white arrow. b: The secondary electron scanning electron microscopy image at E0 ⫽ 5 keV presents a top-view image of the same wedge sample with indicated wedge angle.
Figure 6. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities of the amorphous C wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 17.28 and t0 ⫽ 70 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections.
Black lines indicate IHAADF for 15 and 30 keV in Figures 8a and 8b, respectively, which are compared with SR-CS-based and M-CS-based MC simulations. An intensity maximum is observed for 15 keV at a sample thickness of ;160 to 200 nm ~depending on the cross-section type!. The reduction of HAADF intensity beyond the maximum results from the fact that electrons are increasingly scattered in very large angles, which cannot be collected by the HAADF detector anymore. The maximum does not yet appear at 30 keV, where an almost linear correlation exists between IHAADF and the sample thickness. A good fit of the experimental data at 30 keV ~Fig. 8b! is obtained for M-CSbased simulations for a wedge with t0 ⫽ 59 nm and a ⫽ 25.08. These parameters correspond well to the data obtained from the top-view SE image ~t0 ⫽ 60 6 15 nm and a ⫽ 26 6 18!. The deviation between experimental and simulated data at t ⱕ 170 nm can be attributed to a change of the wedge angle, which is also visible in Figure 7b. The
HAADF STEM intensity increases almost linearly up to a sample thickness of 400 nm and allows reliable thickness quantification. However, the fit at 15 keV is not satisfactory for any of the two cross-sections. The shape of the experimental curve is reproduced by simulations based on M-CSs but IHAADF is substantially lower than predicted, which will be considered further in the Discussion section. A 30 keV HAADF STEM cross-section image of the GaN wedge sample is shown in Figure 9a and a top-view SE SEM image of the same wedge in Figure 9b. An additional C protection layer with dark contrast is located between the Pt protection layer and GaN in Figure 9a. Experimental and simulated HAADF STEM intensities as a function of the sample thickness are presented for 15 and 30 keV in Figures 10a and 10b, respectively. Maxima of the HAADF STEM intensity now occur for both electron energies. Good agreement is found between the experimental data and M-CSbased simulations for 30 keV for a fit with t0 ⫽ 60 nm and
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Figure 7. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of the Si wedge sample with the position of the intensity linescan marked by a white arrow. b: 30 keV secondary electron scanning electron microscopy top-view image of same sample with indicated wedge angle.
Figure 8. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities for a Si wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 25.08 and t0 ⫽ 59 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections.
a ⫽ 29.08. The latter values are in good agreement with the parameters derived from Figure 9b ~t0 ⫽ 58 6 15 nm and a ⫽ 30 6 18!. In analogy to C and Si, a substantial disagreement is observed between experimental and simulated IHAADF at 15 keV independent of the type of scattering cross-section. The apparent agreement of IHAADF with the SR-CS-based simulated curve for larger thicknesses is considered to be accidental. As in the case of Si, the shape of the curve and the position of the maximum correspond to the M-CS-based simulation—apart from the fact that the experimental data is displaced toward lower intensities. The fact that intensity maxima occur in the considered thickness range yields ambiguous sample thickness values. This problem can be solved by measurements at two different primary energies. The procedure is illustrated by lightgray dashed lines in Figures 10a and 10b. For IHAADF ⫽ 0.5 and E0 ⫽ 30 keV at the sample position of interest, local sample thickness values of 130 or 290 nm ~Fig. 10b! would result, which are given by the intersections of the horizontal
line with the calibration curve. The measurement is then repeated at E0 ⫽ 15 keV, which yields clearly different IHAADF values at t-values ~see Fig. 10a!, which allow the determination of the true sample thickness. Tungsten was studied as a representative of a high-Z material. A 30 keV HAADF STEM cross-section image of a W wedge sample is presented in Figure 11a and a top-view SE SEM image in Figure 11b. Slight intensity variations close to the thin edge of the wedge in Figure 11a indicate that the sample is polycrystalline. These grains are etched differently by the Ga⫹-ion beam during FIB preparation, which may induce a rough wedge surface. Although the grains show slightly different contrast, this did not seriously affect the intensity measurement because the contrast change from one grain to the other is negligible compared with the contrast change because of the increasing wedge thickness. Experimental and simulated IHAADF values as a function of the sample thickness are presented for 15 and 30 keV in Figures 12a and 12b,
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Figure 9. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of the GaN wedge sample with the position of the intensity linescan marked by a white arrow. b: 30 keV secondary electron scanning electron microscopy top-view image of same sample with indicated wedge angle.
Figure 10. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities for a GaN wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 29.08 and t0 ⫽ 60 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections. The horizontal and vertical light-gray dashed lines illustrate the solution of possible thickness ambiguities as described in the text.
respectively. The surface roughness induces slight undulations in the 30 keV linescan ~Fig. 12b!. This effect is less pronounced in Figure 12a at 15 keV because of the reduced spatial resolution. Good agreement is observed between experimental data and simulations based on M-CSs for t0 ⫽ 35 nm and a ⫽ 17.08. These values correspond well with measurements in the top-view SE image ~t0 ⫽ 40 6 15 nm and a ⫽ 18 6 18!. We note that the maximum of IHAADF,s occurs at rather low thickness values t ⫽ 20 nm for E0 ⫽ 15 keV and t ⫽ 50 nm for E0 ⫽ 30 keV. Thickness measurements will therefore be performed on the descending branch of the IHAADF,s versus t-curve for a large interval of sample thicknesses. On the other hand, the steep slope of the curves at small t-values will allow thickness measurements with high accuracy. The procedure of sample thickness determination is illustrated for a W sample with plan-parallel surfaces and a priori unknown thickness in the following. Figure 13a shows
a 30 keV HAADF STEM cross-section image of the W sample with a Pt protection layer on top. The same region was imaged at 15 keV. The corresponding top-view SE SEM image is shown in Figure 13b, confirming that the sample is plane parallel in this region. IHAADF was measured for both energies in the area marked in Figure 13a yielding I30 keV ⫽ 0.316 6 0.005 and I15 keV ⫽ 0.075 6 0.004. These values are indicated as horizontal arrows with error margins ~dashed lines! in the IHAADF,s versus t-curves based on M-CSs for E0 ⫽ 15 and 30 keV in Figures 13c and 13d ~same curves as Figs. 12a and 12b!. The intersections of the horizontal arrows with the simulated curves yield possible thickness values indicated by vertical arrows: t130 keV ⫽ 17 6 2 nm, t230 keV ⫽ 151 6 3 nm, t115 keV ⫽ 2 6 2 nm, and t215 keV ⫽ 167 6 3 nm. As 2 nm is an implausible sample thickness, this value can be discarded. The correct thickness is given by t2 with an average value tHAADF ⫽ 159 6 10 nm, taking the error margins of the single measurements into account. The
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Figure 11. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of the W wedge sample with the position of the intensity linescan marked by a white arrow. b: 5 keV secondary electron scanning electron microscopy top-view image of the same sample with indicated wedge angle.
Figure 12. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities for a W wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 17.08 and t0 ⫽ 35 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections.
determined thickness is confirmed in the top-view SE image, which reveals a thickness of 155 6 15 nm.
D ISCUSSION It is shown in the previous section that the thickness of TEM samples with known composition can be well quantified by comparing MC simulations and HAADF STEM intensities if suitable imaging parameters and scattering cross-sections are chosen. Favorable conditions are achieved for a steep slope of the IHAADF versus t-curve. Conditions close to the maximum IHAADF should be avoided where IHAADF does not sensitively depend on the sample thickness. To identify unfavorable conditions the thickness at maximum IHAADF was calculated for a broad selection of pure elements at E0 ⫽ 15, 20, 25, and 30 keV. Figure 14a shows that the thickness at maximum IHAADF oscillates strongly with increasing Z because, apart from Z, the material den-
sity also strongly influences IHAADF . Large thicknesses at maximum IHAADF are obtained for low-density materials like Na, Ca, or Ba even at high atomic numbers. The oscillations disappear if the thickness values at maximum IHAADF are multiplied by the material density as shown in Figure 14b. The curves shown in Figure 14 serve as a guideline for selecting suitable E0 values. The primary electron energy should be chosen that the maximum acceptable ~or expected! sample thickness significantly lies below the curves in Figures 14a and 14b, where measurements are performed on the ascending section of the IHAADF versus t-curve. Since the thickness at maximum IHAADF becomes small for heavy elements this will not always be possible and analyses have to be performed on the descending section of the IHAADF versus t-curves. The precision will be reduced because of the smaller gradient compared to the ascending part of the curve. However, Figures 12a and 12b for W and Figure 10b
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Figure 13. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of a plane parallel region of the W sample with the position of the intensity measurement marked by a white box and ~b! 30 keV secondary electron scanning electron microscopy top-view image of same sample. Comparison of the measured intensity of a plane parallel W sample region with simulated intensities based on M-CSs for ~c! E0 ⫽ 15 keV, and ~d! E0 ⫽ 30 keV. The horizontal arrows indicate the measured intensity values. The vertical arrows give the resulting thickness values. The surrounding dashed lines illustrate the error margins of the measured intensity values and resulting thickness values.
Figure 14. Overview of ~a! the sample thickness at maximum IHAADF and ~b! mass thickness at maximum IHAADF as a function of the atomic number for pure elements at E0 ⫽ 15, 20, 25, and 25 keV derived from Monte Carlo simulations using Mott cross-sections.
for GaN show that rather steep gradients for the descending section of the IHAADF versus t-curves allow thickness measurements with rather good accuracy. Although conditions should be generally avoided where the sample thickness corresponds to the maximum of the IHAADF versus t-curve, the maximum can be exploited in some cases for thickness determination. This was shown by Pfaff et al. ~2011! who studied thin carbon and polymer films with rather homogeneous thickness. In this case, the primary electron energy for contrast reversal was deter-
mined, which corresponds to the maximum of the IHAADF versus t-curve. Either MC simulations or a suitable empirical equation, which relates the thickness at maximum IHAADF and E0 , can then be used to determine the sample thickness. Another consequence of the maximum of the IHAADF versus t-curves is an ambiguity regarding the evaluated thicknesses if preinformation on the relevant sample thickness range is not available. Information on the relevant sample thickness range is usually available if TEM samples
Sample Thickness Determination by Low-Energy STEM
are prepared by FIB milling. Without preinformation, the ambiguity can be resolved if measurements are performed for at least two different E0 values. This procedure was illustrated in the previous section for GaN and W. For evaluation of the precision of the technique the errors for IHAADF and IHAADF,s have to be considered. The experimental HAADF STEM intensities are affected by the statistical error and thickness variations because of averaging over a sample region where the latter is not an intrinsic error of the technique. The statistical error can be estimated by the M N/N criterion ~N: number of electrons in the HAADF range!. The used pixel dwell time of 30 ms and beam current of 0.62 nA give a dose of 10 6 electrons/ pixel in case of full illumination. From the simulations and average gray levels of 10,000–30,000 we estimate that 15,000– 60,000 electrons are scattered in the HAADF range. This results in a statistical error ⱕ1% and, of course, also affects the determination of IB and IW required for normalization. Gaussian error propagation yields in unfavorable cases statistical errors up to 5% for IHAADF . DIHAADF leads to an error for the sample thickness Dt, which is determined by the slope S ⫽ DI/Dt of the IHAADF,s versus t-curve. For the examples in the preceding section, it varies from 0.001 nm⫺1 ~Figs. 4b and 4c at E0 ⫽ 30 keV around t ⫽ 100 nm! to 0.01 nm⫺1 ~Fig. 12b, W at E0 ⫽ 30 keV around t ⫽ 30 nm!. The slope determines the error for the evaluated sample thickness Dt of 1–3%. Absolute values can be inferred from the thickness determination of the plane parallel W sample in the previous section. These values are significantly affected by real thickness variations of the analyzed sample region. We note that the sensitivity of low-energy HAADF STEM is high for FHBC at 15 keV with respect to small thickness changes. In Figure 4a, the error for the nominal thickness of a multiply folded film could be easily detected. Evaluation of the precision of the technique also requires the consideration of possible errors in the MC simulations. The statistical error, given by the M N/N criterion, can be neglected if the simulations are performed for a sufficiently large number of electrons. Our simulations are performed with 10 6 incident electrons, which yields N values between 10 5 and 5 ⫻ 10 5 electrons in the HAADF angular range and a statistical error ,0.3%. The measured working distance and radii of the HAADF segment have to be considered as sources for systematical errors. They determine the integration limits for evaluation of the MC simulations, as illustrated by vertical lines in Figure 2. The influence of these parameters depends on the shape of the scattering angle distribution, which is determined by the sample composition, thickness, and primary electron energy. The sum of systematical and statistical error for IHAADF,s is estimated to be below 0.5%. Larger errors can result if inadequate scattering crosssections are chosen for the MC simulations, which need to be carefully selected according to E0 and Z. SR-CSs describe the HAADF STEM intensity of low-Z materials like FHBC for E0 ⫽ 15 and 30 keV and for carbon at 30 keV. M-CSbased MC simulations yield good agreement between IHAADF
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and IHAADF,s at 30 keV for materials with intermediate and high atomic numbers, like Z ⫽ 14 ~silicon!, Z ⫽ 19 ~GaN!, and Z ⫽ 74 ~tungsten!. However, the measured intensities are consistently lower than IHAADF,s at 15 keV for carbon, silicon, and GaN although the shape of the curve and the position of maximum IHAADF resemble M-CS-based simulations for Si and GaN. We note that the discrepancy between IHAADF and IHAADF,s increases with decreasing E0 , which was observed from additional measurements at E0 ⫽ 20 and 25 keV ~not presented here!. We give two possible explanations for this discrepancy. On the one hand, the scattering cross-sections might be not applicable for the given Z and E0 values. Since the effect appears for SR-CSs ~carbon! as well as for M-CSs ~silicon and GaN!, this would indicate that neither cross-section model is suited to adequately describe IHAADF at 15 keV for materials with low and intermediate atomic numbers. On the other hand, the effect could result from yet unaddressed errors in the description of detector-related effects for which MC calculations need to be corrected. Although regarded as constant, the efficiency « and backscattering coefficient h of the detector could be energy dependent leading to exaggerated IHAADF,s at 15 keV. This hypothesis was tested by widely varying both parameters. However, the resulting effect on the IHAADF,s versus t-curve was too small to explain the deviations. The introduction of a further correction factor could formally solve the problem but lacks physical justification. Further investigations on this effect are needed. Depending on the atomic number of the sample material and E0 it is suggested that simulated and experimental HAADF STEM intensities are compared using calibration curves from reference samples with known thicknesses like the wedge samples in this work. Our results eventually shed some light on general recommendations regarding the use of SR-CSs and M-CSs because normalized IHAADF data can be used to evaluate the validity range of scattering cross-sections in MC simulations. Statements in the literature are rather vague regarding conditions for the applicability of scattering cross-sections. Browning et al. ~1994! stated that SR-CSs are sufficient for low-Z materials and high energies, while M-CSs have to be used for high-Z materials and intermediate energies ~0.1– 30 keV!. Shimizu and Ding ~1992! are more specific and suggest the use of SR-CSs in the energy range of 1–20 keV for elements lighter than aluminum ~Z ⫽ 13!. Our results in the energy range of 15–30 keV partially confirm this recommendation because SR-CSs well describe the experimental data upto Z ⫽ 6 while M-CSs are adequate for Z ⱖ 14 at 30 keV. Further experiments with suitable samples could help to define the regimes for SR-CSs and M-CSs more precisely and analyze the discrepancies between IHAADF and IHAADF,s at 15 keV. According to Czyzewski et al. ~1990!, small differences in the scattering cross-sections only cause little effects in MC simulations because of the averaging effect of the statistical scattering processes. With reference to Reimer ~1998! they state that SR-CSs and M-CSs are approximately
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equal for low-Z materials at primary energies E0 ⱖ 1 keV for scattering angles u ⱖ 108. In contrast to the suggestion of Czyzewski et al. ~1990!, IHAADF,s based on M-CSs and SRCSs differ significantly in our study for the investigated energy range. Kotera et al. ~1981! analyzed MC simulations up to 10 kV for gold and found that the SR-CSs underestimate the scattering into the HAADF angular range. This effect is confirmed in our simulations for sample thicknesses larger than the thickness at maximum IHAADF,s where the simulated intensities are generally higher if M-CSs are applied. However, in the lower thickness range before the maximum the opposite is observed, where SR-CSs give higher intensities than M-CSs. Furthermore, the intensity maximum of SR-CS-based simulations is generally higher compared to M-CS-based simulations and shifted toward lower thicknesses. Finally, different approaches exist for the calculation of M-CSs and SC-CSs in addition to the cross-sections used for the MC simulations in this work, for example, by Murata et al. ~1971!, Reimer and Lödding ~1984!, Gauvin and Drouin ~1993!, Browning et al. ~1994!, or Salvat et al. ~2005!, which need to be systematically evaluated.
HAADF STEM intensities were found at 15 keV for materials with Z-values between 6 ~carbon! and 19 ~GaN!, independent on the chosen model for the scattering cross-section. The source of this discrepancy needs further investigations. It could be related to the scattering cross-sections themselves or a yet unaddressed error in the description of detector-related effects. The accuracy of the method increases with the slope of the IHAADF versus t-curve. Accordingly, conditions should be avoided where measurements are performed close to the intensity maximum of the IHAADF versus t-curve where the slope is small. A guideline for the selection of suitable E0 values was given by calculating the thickness or mass-thickness, respectively, at maximum IHAADF as a function of Z and E0 . The main source of error can be the simulated intensities if unfavorable conditions are chosen ~scattering cross-sections, primary electron energies!. This error can be excluded if IHAADF versus t calibration curves are acquired using TEM samples with a known thickness profile like the wedge samples in this work. When choosing proper conditions, the overall accuracy of the method is better than 65% if the sample thickness is homogeneous in the analyzed region.
S UMMARY
A CKNOWLEDGMENTS
A method for thickness determination for samples with known composition is presented in this work. The procedure is based on HAADF STEM intensity measurements at primary energies E0 ⱕ 30 keV, which can be performed in an SEM equipped with a STEM detector. The normalized measured intensity IHAADF is directly compared with calculated intensities based on MC simulations. Adequate adjustment of brightness and contrast ~amplification! is required and the MC simulations need to be corrected for detectorrelated effects. The IHAADF versus t-curves are characterized by an intensity maximum at a thickness, which depends on the atomic number of the sample material and the electron energy. This can lead to ambiguous values for the sample thickness, which can be resolved by performing measurements for two different primary electron energies. The procedure was explicitly demonstrated for GaN and a W sample with a priori unknown thickness. Five samples covering a wide range of ~average! atomic numbers between 3.5 ~FHBC! to 74 ~tungsten! were investigated. Either thin films with homogeneous thickness or wedge-shaped samples with a well-defined thickness profile were studied. Experimental and calculated HAADF STEM intensities were compared at 15 and 30 keV. The MC simulations were performed with SR-CSs and M-CSs. M-CSs are a good choice at 30 keV for materials with intermediate and high atomic numbers as verified for Z ⫽ 14 ~silicon!, 19 ~GaN!, and 74 ~tungsten!. The choice of the scattering cross-section is more complex for light-weight materials. FHBC ~Z ⫽ 3.5! at 15 and 30 keV as well as carbon at 30 keV can be described well by MC simulations based on SR-CSs. Discrepancies between measured and simulated
We thank M.F.G. Klein @Light Technology Institute ~LTI!, Karlsruhe Institute of Technology# for fabrication of the FHBC films, Dr. D. Cooper ~CEA-Leti, Grenoble! for providing the SiGe sample, and Dr. D. Hu and P. Ganz @Center for Functional Nanostructures ~CFN!, Karlsruhe Institute of Technology# for providing the GaN wafer. We also acknowledge project funding by the German Research Foundation ~DFG!.
R EFER ENCES Bishop, H. ~1967!. Electron scattering in thick targets. Br J Appl Phys 18, 703–715. Browning, R., Li, T., Chui, B., Ye, J., Pease, R., Czyzewski, Z. & Joy, D. ~1994!. Empirical forms for the electron/atom elastic scattering cross sections from 0.1 to 30 keV. J Appl Phys 76, 2016–2022. Czyzewski, Z., MacCallum, D., Romig, A. & Joy, D. ~1990!. Calculations of Mott scattering cross section. J Appl Phys 68, 3066–3072. Egerton, R. ~1986!. Electron Energy-Loss Spectroscopy in the Electron Microscope. New York: Plenum Press. Gauvin, R. & Drouin, D. ~1993!. A formula to compute total elastic Mott cross-sections. Scanning 15, 140–150. Heinrich, K. ~1981!. Electron Beam X-Ray Microanalysis. New York: Van Nostrand Reinhold Company. Joy, D. & Luo, S. ~1989!. An empirical stopping power relationship for low-energy electrons. Scanning 11, 176–180. Klein, T., Buhr, E. & Frase, C.G. ~2012!. TSEM: A review of scanning electron microscopy in transmission mode and its applications. In Advances in Imaging and Electron Physics, Hawkes, P.W. ~Ed.!, pp. 297–356. New York: Elsevier. Kotera, M., Murata, K. & Nagami, K. ~1981!. Monte Carlo simulation of 1–10 keV electron scattering in a gold target. J Appl Phys 52, 997–1003.
Sample Thickness Determination by Low-Energy STEM Krzyzanek, V. & Reichelt, R. ~2008!. Mass thickness determination of thin specimens using high-resolution scanning electron microscopy. EMC 2008, 14th European Microscopy Congress, Aachen, September 1–5, 2008. Lehmann, M. & Lichte, H. ~2002!. Tutorial on off-axis electron holography. Microsc Microanal 8, 447–466. Morandi, V. & Merli, P. ~2007!. Contrast and resolution versus specimen thickness in low energy scanning transmission electron microscopy. J Appl Phys 101, 114917. Mott, N.F. & Massey, H.S.W. ~1949!. The Theory of Atomic Collisions. Oxford, UK: Clarendon Press. Motz, J., Olsen, H. & Koch, H. ~1964!. Electron scattering without atomic or nuclear excitation. Rev Mod Phys 36, 881–928. Murata, K., Matsukawa, T. & Shimizu, R. ~1971!. Monte Carlo calculations on electron scattering in a solid target. Jpn J Appl Phys 10, 678–686. Pfaff, M., Müller, E., Klein, M.F.G., Colsmann, A., Lemmer, U., Krzyzanek, V., Reichelt, R. & Gerthsen, D. ~2011!. Low-energy electron scattering in carbon-based materials analyzed by scanning transmission electron microscopy and its application to sample thickness determination. J Microsc 243, 31–39. Powell, C., Jablonski, A. & Salvat, F. ~2005!. NIST databases with electron elastic-scattering cross sections, inelastic mean free paths, and effective attenuation lengths. Surf Interface Anal 37, 1068–1071. Reimer, L. ~1998!. Scanning Electron Microscopy. Heidelberg, Germany: Springer Verlag. Reimer, L. & Lödding, B. ~1984!. Calculation and tabulation of Mott cross-sections for large-angle electron scattering. Scanning 6, 128–151. Ritchie, N. ~2005!. A new Monte Carlo application for complex sample geometries. Surf Interface Anal 37, 1006–1011.
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Rosenauer, A., Gries, K., Müller, K., Pretorius, A., Schowalter, M., Avramescu, A., Engl, K. & Lutgen, S. (2009). Measurement of specimen thickness and composition in AlxGa1⫺xN/GaN using high-angle annular dark field images. Ultramicroscopy 109, 1171–1182. Rutherford, E. ~1911!. The scattering of a- and b-particles by matter and the structure of the atom. Phil Mag 21, 669–688. Salvat, F., Jablonski, A. & Powell, C.J. ~2005!. ELSEPA—Dirac partial-wave calculation of elastic scattering of electrons and positrons by atoms, positive ions and molecules. Comput Phys Commun 165, 157–190. Scholze, F., Henneken, H., Kuschnerus, P., Rabus, H., Richter, M. & Ulm, G. ~2000!. Determination of the electron-hole pair creation energy for semiconductors from the spectral responsivity of photodiodes. Nucl Instrum Meth A 439, 208–215. Shimizu, R. & Ding, Z.-J. ~1992!. Monte Carlo modeling of electron-solid interactions. Rep Prog Phys 55, 487–531. Treacy, M. & Gibson, J. ~1993!. Coherence and multiple scattering in “Z-contrast” images. Ultramicroscopy 52, 31–53. Van den Broek, W., Rosenauer, A., Goris, B., Martinez, G., Bals, S., Van Aert, S. & Van Dyck, D. ~2012!. Correction of non-linear thickness effects in HAADF STEM electron tomography. Ultramicroscopy 116, 8–12. Volkenandt, T., Müller, E., Hu, D., Schaadt, D. & Gerthsen, D. ~2010!. Quantification of sample thickness and Inconcentration of InGaAs quantum wells by transmission measurements in a scanning electron microscope. Microsc Microanal 16, 604–613. Williams, D.B. & Carter, C.B. ~2009!. Transmission Electron Microscopy. New York: Springer.
Microsc. Microanal. 20, 111–123, 2014 doi:10.1017/S1431927613013913
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© MICROSCOPY SOCIETY OF AMERICA 2014
Sample Thickness Determination by Scanning Transmission Electron Microscopy at Low Electron Energies Tobias Volkenandt,* Erich Müller, and Dagmar Gerthsen Laboratorium für Elektronenmikroskopie, Karlsruher Institut für Technologie (KIT), Engesserstr. 7, 76131 Karlsruhe, Germany
Abstract: Sample thickness is a decisive parameter for any quantification of image information and composition in transmission electron microscopy. In this context, we present a method to determine the local sample thickness by scanning transmission electron microscopy at primary energies below 30 keV. The image intensity is measured with respect to the intensity of the incident electron beam and can be directly compared with Monte Carlo simulations. Screened Rutherford and Mott scattering cross-sections are evaluated with respect to fitting experimental data with simulated image intensities as a function of the atomic number of the sample material and primary electron energy. The presented method is tested for sample materials covering a wide range of atomic numbers Z, that is, fluorenyl hexa-peri-hexabenzocoronene ~Z ⫽ 3.5!, carbon ~Z ⫽ 6!, silicon ~Z ⫽ 14!, gallium nitride ~Z ⫽ 19!, and tungsten ~Z ⫽ 74!. Investigations were conducted for two primary energies ~15 and 30 keV! and a sample thickness range between 50 and 400 nm. Key words: scanning transmission electron microscopy, sample thickness determination, Monte Carlo simulation, scattering cross-section, quantification, low energy
I NTR ODUCTION Knowledge of the local sample thickness is essential in transmission electron microscopy ~TEM! for the quantification of any image information. For this reason, several methods have been developed to determine the local TEM sample thickness. Plasmon losses obtained from electron energy loss spectra ~Egerton, 1986!, convergent beam electron diffraction ~Williams & Carter, 2009!, electron holography ~Lehmann & Lichte, 2002!, and thickness contours under well-defined excitation conditions ~Williams & Carter, 2009! can be exploited as well as scanning transmission electron microscopy ~STEM! at TEM-typical energies ~Rosenauer et al., 2009; Van den Broek et al., 2012!. These techniques vary in precision, complexity, and spatial resolution. More recently, STEM in the high-angle annular darkfield ~HAADF! mode at primary energies E0 ⱕ 30 keV was suggested as a promising technique to determine the TEM sample thickness ~Morandi & Merli, 2007; Krzyzanek & Reichelt, 2008; Volkenandt et al., 2010; Pfaff et al., 2011; Klein et al., 2012!. This method is based on the comparison of experimental and calculated HAADF STEM intensities. It can be applied if the composition of the sample material is known. It is advantageous that low-energy STEM images can be recorded quickly in an easy-to-operate SEM fitted with a STEM detector. Combining low-energy STEM in an SEM with a focused ion beam ~FIB! system for TEM sample preparation facilitates thickness quantification directly after FIB sample preparation. Moreover, knock-on damage is negligible because of the low electron energy in an SEM. Spatial resolution is in the range of 1 nm because of the small Received August 1, 2013; accepted November 6, 2013 *Corresponding author. E-mail: [email protected]
interaction volume for thin samples, but decreases with increasing sample thickness because of beam broadening. The quantification of low-energy HAADF STEM data requires an adequate description of the experimental HAADF STEM intensities. Monte Carlo ~MC! simulations are well established for the calculation of the low-energy HAADF STEM intensity because effects of Bragg diffraction can be neglected at these low electron energies and large scattering angles ~Treacy & Gibson, 1993!. However, the results of MC simulations depend on the scattering cross-sections used. Depending on the atomic number of the sample material and the electron energy screened Rutherford cross-sections ~SR-CS! ~Rutherford, 1911; Bishop, 1967! or Mott crosssections ~M-CSs! ~Mott & Massey, 1949! can be used. Different methods were applied to estimate the effects of atomic structure and screening potentials. This leads to a variety of scattering equations, which cannot be solved analytically. Depending on the approximations the equations are only valid for a limited range of parameters ~Motz et al., 1964!. In general, SR-CSs adequately describe electron scattering for low atomic number ~low Z! materials while M-CSs are used for higher atomic numbers. Browning et al. ~1994! stated that SR-CSs give acceptable results for high electron energies and low-Z materials. M-CSs are applied for low to intermediate incident energies ~100 eV–30 keV! and high-Z materials. An estimate on the accuracy is not straightforward. Only comparison with experimental data can confirm the validity of a particular model for the calculation of scattering cross-sections. In this work we focus on the application of low-energy HAADF STEM for TEM sample thickness determination. The measured HAADF STEM intensities are normalized with respect to the intensity of the incident electron beam
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and can thus be directly compared with simulated intensities. For this purpose, wedge-shaped samples with a wellknown thickness profile were prepared by FIB milling. The method is tested for different materials covering a wide range of ~average! atomic numbers between 3.5 @fluorenyl hexa-peri-hexabenzocoronene ~FHBC!# and 74 ~tungsten!. The experimental results are compared with MC simulations on the basis of screened SR-CSs and M-CSs, which yield information on the applicability of the scattering cross-sections as a function of atomic number at electron energies between 15 and 30 keV.
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Experimental Setup Sample preparation and STEM imaging were performed with an FEI Strata 400S dual-beam microscope, which combines an SEM equipped with a Schottky field-emission gun and a focused Ga⫹-ion beam ~FIB! system. It is equipped with an FEI STEM III semiconductor detector positioned below the sample with a fixed distance of 18.8 mm between the objective lens pole piece and detector. The detector is divided into six ring-like segments where the outermost ring is the HAADF segment. The scattering angle range is determined by the inner and outer radii of the selected segment and the working distance between the sample and the objective lens pole piece. The latter was chosen as 6.5 mm for this work. Accordingly, electrons transmitted into a hollow cone between 187 and 683 mrad are detected by the HAADF segment. Primary energies of 15, 20, 25, and 30 keV were used for imaging to detect possible energy-related effects. Only the HAADF ring was used in this work. However, the presented approach can be also applied for the bright-field or annular dark-field ~ADF! segments covering smaller scattering angle ranges if conditions leading to Bragg diffraction are avoided ~e.g., by tilting!. In our case, the signals of the ADF segments were too low for the studied sample thickness range because of the small detection area of these segments. During imaging special attention has to be paid to the contrast ~gain! and brightness ~offset! settings, which are usually tuned to achieve optimum image contrast. While this is useful for enhancing the image contrast it is unfavorable for this work because the image intensity must be quantitatively comparable with MC simulations. To ensure comparability with the simulations, the intensity is normalized with respect to the incident beam. For normalization of the measured intensities we use the following procedure. The detector is first directly illuminated without sample to measure the intensity of the incident electron beam. Contrast and brightness are then adjusted such that the inactive bright-field and ADF detector segments appear black, that is, they are assigned with a gray level IB close to 0. In contrast, the active HAADF segment appears white, that is, it is assigned with a gray level IW close to 65,535 for our 16-bit detector. Oversaturation and undersaturation have to be carefully avoided during this procedure. Figure 1 shows such a detector reference
Figure 1. High-angle annular dark-field scanning transmission electron microscopy ~HAADF-STEM! image of the STEM detector. The active HAADF segments show up bright while all other segments appear dark. The marked areas illustrate the regions that were used to measure the black and white levels of the detector.
image, which was taken at the lowest possible magnification. Only a small, but for the contrast adjustment sufficient innermost part of the HAADF segment can be seen. Additionally, the sample holder ~which was retracted for this image! usually obstructs the upper half of the reference image. That is why the black and white levels are measured in the regions in the lower half of the image as marked in Figure 1. The detector settings remain unchanged for the subsequent images of the samples. The samples typically appear at intermediate gray levels, of about 30,000, depending on the sample thickness and primary electron energy. The intensity values of the sample IS are measured and subtracted by the black level IB measured before in the detector reference image. This offset correction is also performed for the measured white level IW . The normalized HAADF STEM intensity of the sample is denoted by IHAADF and given by the following equation: IHAADF ⫽
IS ⫺ IB IW ⫺ IB
.
~1!
IHAADF can be directly compared with MC simulations. Contrast tuning of the detector with respect to IB and IW is crucial for the intensity measurements in the following because it effectively allows an absolute measurement of the intensity. But this only holds if the amplification characteristic of the detector is linear. Therefore, this was checked by imaging the detector using a large range of beam currents and all primary electron energies of interest. Brightness and contrast were adjusted for the highest beam current and then kept constant. The image intensity of the selected detector segment was investigated as a function of the beam current, which was measured by a Faraday cup. A linear behavior was found between 10 pA and 10 nA, which confirms the linear response of the detector.
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To achieve agreement between measured and simulated intensities, the simulated HAADF intensities were corrected by taking into account the following detector-related effects. According to Reimer ~1998!, the signal of a semiconductor STEM detector in terms of the collection current ICC can be described by the following equation: ICC ⫽ N~1 ⫺ h!
E ⫺ Eoff Ehp
«,
~2!
with the number of electrons N impinging on the detector, the backscatter coefficient of the detector material h, the charge collection efficiency «, the average energy of the scattered electrons E, and the energy offset Eoff , which takes into account the energy loss of electrons while entering the detection layer. The energy for generating an electron-hole pair within the detection layer is denoted by Ehp . For the silicon-based STEM detector in this work we used « ⫽ 1 ~Reimer, 1998!, Ehp ⫽ 3.6 eV ~Scholze et al., 2000! and Eoff ⫽ 3 keV ~determined experimentally!, as confirmed in a previous study ~Volkenandt et al., 2010!. The backscatter coefficient h was calculated according to the following equation ~Reimer, 1998!: h ⫽ ~1 ⫹ cos u!⫺9 / M Z,
A0 ⫺ Ab A0
,
~4!
with the area of the complete HAADF ring A0 and the insensitive detector area Ab . The area of the complete HAADF ring is given by the inner and outer radii rmax and rmin of the detector in the following equation: 2 2 A 0 ⫽ p~rmax ⫺ rmin !,
~5!
Ab is determined by nb ⫽ 6 blind gaps with a width bb ⫽ 0.078 mm in the following equation: A b ⫽ nb bb ~rmax ⫺ rmin !.
IHAADF, s ⫽
~3!
with the scattering angle u and the atomic number Z ⫽ 14 for the detector material. In addition, a geometric effect has to be considered. The ring-like HAADF segment is divided into six subsections ~see Fig. 1!. Owing to this segmentation, there are insensitive regions on the detector, which do not contribute to the HAADF STEM signal. The real detection area is described by a geometric correction factor given in the following equation: cA ⫽
Figure 2. Simulated scattering angle distribution for C ~dashed line! and Pt ~solid line! films of 150 nm thickness at 15 keV primary energy. The gray vertical lines mark the angular range covered by the high-angle annular dark-field segment of the scanning transmission electron microscopy detector.
~6!
The corrections described by equations ~2–6! are applied during the integration of the scattering angle histograms obtained from the MC simulations. This yields the corrected simulated intensity on the HAADF segment Jcorr , which has to be normalized with respect to the incident intensity J0 for comparison with measured intensities. J0 is given by the number of incident electrons 10 6 corrected for the physical detector effects. Energy losses are not considered for J0 so that E equals E0 in equation ~2!. The simulated HAADF intensity IHAADF,s is then given by
Jcorr J0
,
~7!
which can be directly compared with the measured HAADF intensity IHAADF if the image was recorded under the abovementioned conditions for absolute intensity measurements.
MC Simulations MC simulations are well established for the calculation of SEM and STEM intensities at electron energies E0 ⱕ 30 keV. Multiple scattering prevails even in thin samples resulting from mean free path lengths in the order of 10 nm and below. MC simulations adequately describe the stochastic nature of the scattering process and are therefore widely used to predict low-energy STEM intensities. In this work we used the NISTMonte package ~Ritchie, 2005! with implemented SR-CSs ~Heinrich, 1981! and M-CSs ~Powell et al., 2005!. The energy loss of the electrons is calculated by the Joy–Luo formalism ~Joy & Luo, 1989!, which is a modification of Bethe’s continuous slowing down approximation. The samples in the simulations are modeled as an infinite, homogeneous film, with constant thickness. Simulation series for different film thicknesses and primary electron energies were performed. The number of incident electrons was chosen to be 10 6 to minimize the statistical error. The simulations yield the scattering angle and energy histogram of the scattered electrons. A typical scattering angle distribution is shown in Figure 2 for carbon and platinum with 150 nm sample thickness for 15 keV. The scattering angle range covered by the STEM detector is marked by vertical lines. As carbon is a weakly scattering material the angular distribution in Figure 2 shows a rather pronounced maximum at small scattering angles around 100 mrad. Platinum on the other hand scatters strongly because of its large atomic number. The angular distribution shows a small but
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broad maximum around 700 mrad. The more pronounced maximum at 2.4 rad stems from backscattered electrons whose number already exceeds the number of electrons, which are transmitted through the sample at 150 nm thickness. This is also recognized in the HAADF signal, which corresponds to the area under the curves between the vertical lines. IHAADF for carbon is larger resulting in brighter image contrast. An increase in thickness would cause the carbon curve to broaden and shift to higher scattering angles reaching a shape comparable to the actual platinum curve at large sample thicknesses. A decrease in thickness would cause the curves to sharpen and shift to lower scattering angles.
Sample Materials The samples were chosen to cover a wide range of atomic numbers Z to show the broad applicability of the method. The sample material with the lowest Z is FHBC, a polymer with the chemical formula C100H98 , an average atomic number of Z ⫽ 3.5, and density r ⫽ 1.1 g/cm 3. Five films with different thicknesses were prepared by spin coating. To determine the film thicknesses with an alternative method, a Dektak profilometer was used. These thickness values will be denoted as nominal thickness values. The precision of the Dektak thickness measurements is 615 nm, given by the standard deviation of the mean value after multiple measurements. Carbon ~Z ⫽ 6, r ⫽ 1.9 g/cm 3 ! was chosen as another low-Z sample material, which was deposited by electronbeam vapor deposition on a mica substrate as an amorphous layer. Using FIB-milling a cross-section lamella was prepared. Electron beam and ion beam induced Pt deposition was used for sample protection. The lamella was thinned to a thickness of about 1 mm, cut free, and transferred to a dedicated copper sample grid using an Omniprobe manipulator needle. The lamella was fixed by ion beam-induced Pt deposition. The lamella was further thinned in a wedgelike shape with a nominal wedge angle of 208 to obtain a well-defined thickness profile. The same FIB-based preparation technique was used to prepare wedge-shaped lamellae of Si ~Z ⫽ 14, r ⫽ 2.3 g/ cm 3 ! and GaN ~average atomic number Z ⫽ 19, r ⫽ 6.1 g/cm 3 !, which can both be considered as representatives of materials with intermediate atomic numbers. The silicon sample is a Si/Ge heterostructure, which was grown by reduced pressure chemical vapor deposition on a Si substrate. It contains four layers with varying Ge concentration embedded in the Si matrix. We did not study the Si/Ge layers because their Ge concentration was not known precisely enough. Only the Si matrix was analyzed in this work. The GaN sample was prepared from a commercial GaN wafer grown by hybrid vapor phase epitaxy. FIB preparation was performed using a nominal wedge angle of 308 for the Si and GaN samples. Tungsten ~Z ⫽ 74, r ⫽ 19.3 g/cm 3 ! was chosen as a representative of heavy materials. The preparation was again performed by FIB with a wedge angle of 208. All samples
Figure 3. 15 keV high-angle annular dark-field scanning transmission electron microscopy image of a fluorenyl hexa-perihexabenzocoronene film with a nominal thickness of 65 nm. White squares mark positions for intensity measurements at different film thicknesses.
were imaged at electron energies of 15, 20, 25, and 30 keV to check energy-dependent effects on the image intensity but only the results obtained at 15 and 30 keV are presented here.
E XPERIMENTAL R ESULTS Figure 3 shows a representative HAADF STEM image of one FHBC film ~Nr. 1! taken at 15 keV. The film does not lie flat on the Cu grid but turned out to be multiply folded. Regions with the lowest intensity correspond to a singlelayer film ~marked by 1 in Fig. 3! while folded regions can be recognized by their higher intensity ~2–5 indicate multiplelayer film regions in Fig. 3!. The folded film regions yield HAADF intensities at integer values of the nominal film thickness of film Nr. 1 of 65 nm. The thickness homogeneity of the FHBC film can be assessed by the homogeneous HAADF STEM intensity. Further, IHAADF values could be obtained from four more FHBC films ~Nr. 2–5! with nominal thicknesses of 90, 106, 121, and 244 nm. Figure 4 depicts IHAADF of all five FHBC films as a function of the nominal thickness for E0 ⫽ 15 keV ~Fig. 4a! and E0 ⫽ 30 keV ~Fig. 4b!. Solid symbols mark single-film IHAADF data ~for all five films!, while empty symbols indicate measurements at different positions of multiply folded film regions ~for films Nr. 1–3!. The dash-dotted line gives IHAADF,s on the basis of M-CSs, while the dashed line shows simulation results using SR-CSs. The experimental results for 15 keV ~Fig. 4a! can be well described by MC simulations based on SR-CSs. This does not seem to apply for the data points of film Nr. 2 ~diamond symbols in Fig. 4!, which, on first sight, can be better fitted with IHAADF,s based on M-CSs. This disagreement can be explained by the error for the nominal film thickness of film Nr. 2 of 90 nm and the error of the profilometer measurements of at least 615 nm. Assuming a smaller nominal thickness of 70 nm
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Figure 4. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities of five fluorenyl hexa-peri-hexabenzocoronene films as a function of the nominal film thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. Solid symbols mark measurements of single-layer film regions, empty symbols mark measurements at positions with multiple-layer film thickness because of folding of the film. The arrows in ~a! mark thickness-corrected IHAADF for film Nr. 2 as described in the text.
results in good agreement with the SR-CS-based simulations as indicated by arrows in Figure 4a. The results on this film demonstrate that the precision of HAADF STEM based measurements is better than the profilometer data. Figure 4b shows data for 30 keV with low intensities and a shallow slope of the IHAADF versus thickness curve. The experimental data can be well fitted by SR-CS-based simulations but discrepancies are observed for larger sample thicknesses. Clearly, lower electron energies are preferred for low-Z materials with respect to the accuracy of thickness determination. Figure 5a shows a 15 keV HAADF STEM cross-section image of the C wedge sample. The mica substrate, the C layer and the Pt protection layer can be well distinguished because of their different intensities. The sample thickness increases from the left to the right because of the wedge geometry along the white arrow, which correlates with the increase of IHAADF . At the thin edge a higher intensity of Pt is observed, as expected, but a contrast inversion occurs for Pt with increasing sample thickness. This effect can be explained by considering Figure 2 because fewer electrons are scattered in the HAADF angular range because of the large fraction of backscattered electrons. It is noted that the Pt protection layer consists of Pt and a considerable fraction of C, which prevents the quantification of the sample thickness in this case. A top-view secondary electron ~SE! SEM image of the same wedge is presented in Figure 5b, which reveals a wedge angle of a ⫽ 18 6 18 and a thickness offset at the thin edge of the wedge t0 ⫽ 50 6 15 nm. Only the capping Pt protection layer is visible in top-view. The sample material below may exhibit different sputtering rates during the FIB treatment, which in effect leads to uncertainties of a and t0 determined from SE SEM images. The white arrow in Figure 5a marks an intensity linescan averaged over a width of 20 px along the wedge starting at the edge. Owing to the wedge geometry each point x along the linescan can be
correlated with a local sample thickness t according to the following equation: t ⫽ t0 ⫹ x{tan a.
~8!
This procedure is performed for all wedge samples in the following. Simulated and experimental IHAADF values for carbon are shown in Figures 6a and 6b for 15 and 30 keV, respectively, which include simulations on the basis of M-CSs and SR-CSs. The experimental data ~black solid line! were derived from the intensity linescan with the position coordinate transformed into local sample thickness according to equation ~8!. The two parameters t0 and a were adjusted leading to horizontal shift and stretching of the linescan to achieve an optimum fit with one of the simulated curves. The intensity values of the linescan remain unchanged. The best fit was obtained for MC simulations based on SR-CSs for t0 ⫽ 70 nm and a ⫽ 17.28 for both electron energies. However, some discrepancies remain in particular for E0 ⫽ 15 keV. The measured intensities are too low, although the shape reproduces the SR-CS-simulated curve. This effect will be further considered in the Discussion section. A deviation at thicknesses below 80 nm is visible for both energies. It is caused by a blunting of the wedge edge, which can be seen in Figure 5b. Blunting of the edge effectively increases the wedge angle and leads to a steeper reduction of the sample thickness and IHAADF . Figure 7 presents a 30 keV HAADF STEM cross-section image of the Si wedge sample in Figure 7a and a top-view SE SEM image in Figure 7b. The four Si/Ge layers in Figure 7a show brighter contrast than the Si matrix. The layer directly under the Pt protection layer shows brighter contrast because of Ga implantation during the FIB preparation process. The sample thickness increases from left to right and IHAADF was measured in the Si matrix area as indicated by the white arrow.
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Figure 5. a: 15 keV high-angle annular dark-field scanning transmission electron microscopy image of the C wedge sample. The position of the intensity linescan is indicated by a white arrow. b: The secondary electron scanning electron microscopy image at E0 ⫽ 5 keV presents a top-view image of the same wedge sample with indicated wedge angle.
Figure 6. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities of the amorphous C wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 17.28 and t0 ⫽ 70 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections.
Black lines indicate IHAADF for 15 and 30 keV in Figures 8a and 8b, respectively, which are compared with SR-CS-based and M-CS-based MC simulations. An intensity maximum is observed for 15 keV at a sample thickness of ;160 to 200 nm ~depending on the cross-section type!. The reduction of HAADF intensity beyond the maximum results from the fact that electrons are increasingly scattered in very large angles, which cannot be collected by the HAADF detector anymore. The maximum does not yet appear at 30 keV, where an almost linear correlation exists between IHAADF and the sample thickness. A good fit of the experimental data at 30 keV ~Fig. 8b! is obtained for M-CSbased simulations for a wedge with t0 ⫽ 59 nm and a ⫽ 25.08. These parameters correspond well to the data obtained from the top-view SE image ~t0 ⫽ 60 6 15 nm and a ⫽ 26 6 18!. The deviation between experimental and simulated data at t ⱕ 170 nm can be attributed to a change of the wedge angle, which is also visible in Figure 7b. The
HAADF STEM intensity increases almost linearly up to a sample thickness of 400 nm and allows reliable thickness quantification. However, the fit at 15 keV is not satisfactory for any of the two cross-sections. The shape of the experimental curve is reproduced by simulations based on M-CSs but IHAADF is substantially lower than predicted, which will be considered further in the Discussion section. A 30 keV HAADF STEM cross-section image of the GaN wedge sample is shown in Figure 9a and a top-view SE SEM image of the same wedge in Figure 9b. An additional C protection layer with dark contrast is located between the Pt protection layer and GaN in Figure 9a. Experimental and simulated HAADF STEM intensities as a function of the sample thickness are presented for 15 and 30 keV in Figures 10a and 10b, respectively. Maxima of the HAADF STEM intensity now occur for both electron energies. Good agreement is found between the experimental data and M-CSbased simulations for 30 keV for a fit with t0 ⫽ 60 nm and
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Figure 7. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of the Si wedge sample with the position of the intensity linescan marked by a white arrow. b: 30 keV secondary electron scanning electron microscopy top-view image of same sample with indicated wedge angle.
Figure 8. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities for a Si wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 25.08 and t0 ⫽ 59 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections.
a ⫽ 29.08. The latter values are in good agreement with the parameters derived from Figure 9b ~t0 ⫽ 58 6 15 nm and a ⫽ 30 6 18!. In analogy to C and Si, a substantial disagreement is observed between experimental and simulated IHAADF at 15 keV independent of the type of scattering cross-section. The apparent agreement of IHAADF with the SR-CS-based simulated curve for larger thicknesses is considered to be accidental. As in the case of Si, the shape of the curve and the position of the maximum correspond to the M-CS-based simulation—apart from the fact that the experimental data is displaced toward lower intensities. The fact that intensity maxima occur in the considered thickness range yields ambiguous sample thickness values. This problem can be solved by measurements at two different primary energies. The procedure is illustrated by lightgray dashed lines in Figures 10a and 10b. For IHAADF ⫽ 0.5 and E0 ⫽ 30 keV at the sample position of interest, local sample thickness values of 130 or 290 nm ~Fig. 10b! would result, which are given by the intersections of the horizontal
line with the calibration curve. The measurement is then repeated at E0 ⫽ 15 keV, which yields clearly different IHAADF values at t-values ~see Fig. 10a!, which allow the determination of the true sample thickness. Tungsten was studied as a representative of a high-Z material. A 30 keV HAADF STEM cross-section image of a W wedge sample is presented in Figure 11a and a top-view SE SEM image in Figure 11b. Slight intensity variations close to the thin edge of the wedge in Figure 11a indicate that the sample is polycrystalline. These grains are etched differently by the Ga⫹-ion beam during FIB preparation, which may induce a rough wedge surface. Although the grains show slightly different contrast, this did not seriously affect the intensity measurement because the contrast change from one grain to the other is negligible compared with the contrast change because of the increasing wedge thickness. Experimental and simulated IHAADF values as a function of the sample thickness are presented for 15 and 30 keV in Figures 12a and 12b,
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Figure 9. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of the GaN wedge sample with the position of the intensity linescan marked by a white arrow. b: 30 keV secondary electron scanning electron microscopy top-view image of same sample with indicated wedge angle.
Figure 10. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities for a GaN wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 29.08 and t0 ⫽ 60 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections. The horizontal and vertical light-gray dashed lines illustrate the solution of possible thickness ambiguities as described in the text.
respectively. The surface roughness induces slight undulations in the 30 keV linescan ~Fig. 12b!. This effect is less pronounced in Figure 12a at 15 keV because of the reduced spatial resolution. Good agreement is observed between experimental data and simulations based on M-CSs for t0 ⫽ 35 nm and a ⫽ 17.08. These values correspond well with measurements in the top-view SE image ~t0 ⫽ 40 6 15 nm and a ⫽ 18 6 18!. We note that the maximum of IHAADF,s occurs at rather low thickness values t ⫽ 20 nm for E0 ⫽ 15 keV and t ⫽ 50 nm for E0 ⫽ 30 keV. Thickness measurements will therefore be performed on the descending branch of the IHAADF,s versus t-curve for a large interval of sample thicknesses. On the other hand, the steep slope of the curves at small t-values will allow thickness measurements with high accuracy. The procedure of sample thickness determination is illustrated for a W sample with plan-parallel surfaces and a priori unknown thickness in the following. Figure 13a shows
a 30 keV HAADF STEM cross-section image of the W sample with a Pt protection layer on top. The same region was imaged at 15 keV. The corresponding top-view SE SEM image is shown in Figure 13b, confirming that the sample is plane parallel in this region. IHAADF was measured for both energies in the area marked in Figure 13a yielding I30 keV ⫽ 0.316 6 0.005 and I15 keV ⫽ 0.075 6 0.004. These values are indicated as horizontal arrows with error margins ~dashed lines! in the IHAADF,s versus t-curves based on M-CSs for E0 ⫽ 15 and 30 keV in Figures 13c and 13d ~same curves as Figs. 12a and 12b!. The intersections of the horizontal arrows with the simulated curves yield possible thickness values indicated by vertical arrows: t130 keV ⫽ 17 6 2 nm, t230 keV ⫽ 151 6 3 nm, t115 keV ⫽ 2 6 2 nm, and t215 keV ⫽ 167 6 3 nm. As 2 nm is an implausible sample thickness, this value can be discarded. The correct thickness is given by t2 with an average value tHAADF ⫽ 159 6 10 nm, taking the error margins of the single measurements into account. The
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Figure 11. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of the W wedge sample with the position of the intensity linescan marked by a white arrow. b: 5 keV secondary electron scanning electron microscopy top-view image of the same sample with indicated wedge angle.
Figure 12. Comparison of measured and simulated normalized high-angle annular dark-field scanning transmission electron microscopy intensities for a W wedge sample as a function of the wedge thickness for ~a! E0 ⫽ 15 keV and ~b! E0 ⫽ 30 keV. The black line indicates the intensity linescan along the wedge with fit parameters a ⫽ 17.08 and t0 ⫽ 35 nm. Gray lines denote simulated intensities on the basis of Mott ~dash-dotted! and screened Rutherford ~dashed! cross-sections.
determined thickness is confirmed in the top-view SE image, which reveals a thickness of 155 6 15 nm.
D ISCUSSION It is shown in the previous section that the thickness of TEM samples with known composition can be well quantified by comparing MC simulations and HAADF STEM intensities if suitable imaging parameters and scattering cross-sections are chosen. Favorable conditions are achieved for a steep slope of the IHAADF versus t-curve. Conditions close to the maximum IHAADF should be avoided where IHAADF does not sensitively depend on the sample thickness. To identify unfavorable conditions the thickness at maximum IHAADF was calculated for a broad selection of pure elements at E0 ⫽ 15, 20, 25, and 30 keV. Figure 14a shows that the thickness at maximum IHAADF oscillates strongly with increasing Z because, apart from Z, the material den-
sity also strongly influences IHAADF . Large thicknesses at maximum IHAADF are obtained for low-density materials like Na, Ca, or Ba even at high atomic numbers. The oscillations disappear if the thickness values at maximum IHAADF are multiplied by the material density as shown in Figure 14b. The curves shown in Figure 14 serve as a guideline for selecting suitable E0 values. The primary electron energy should be chosen that the maximum acceptable ~or expected! sample thickness significantly lies below the curves in Figures 14a and 14b, where measurements are performed on the ascending section of the IHAADF versus t-curve. Since the thickness at maximum IHAADF becomes small for heavy elements this will not always be possible and analyses have to be performed on the descending section of the IHAADF versus t-curves. The precision will be reduced because of the smaller gradient compared to the ascending part of the curve. However, Figures 12a and 12b for W and Figure 10b
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Figure 13. a: 30 keV cross-section high-angle annular dark-field scanning transmission electron microscopy image of a plane parallel region of the W sample with the position of the intensity measurement marked by a white box and ~b! 30 keV secondary electron scanning electron microscopy top-view image of same sample. Comparison of the measured intensity of a plane parallel W sample region with simulated intensities based on M-CSs for ~c! E0 ⫽ 15 keV, and ~d! E0 ⫽ 30 keV. The horizontal arrows indicate the measured intensity values. The vertical arrows give the resulting thickness values. The surrounding dashed lines illustrate the error margins of the measured intensity values and resulting thickness values.
Figure 14. Overview of ~a! the sample thickness at maximum IHAADF and ~b! mass thickness at maximum IHAADF as a function of the atomic number for pure elements at E0 ⫽ 15, 20, 25, and 25 keV derived from Monte Carlo simulations using Mott cross-sections.
for GaN show that rather steep gradients for the descending section of the IHAADF versus t-curves allow thickness measurements with rather good accuracy. Although conditions should be generally avoided where the sample thickness corresponds to the maximum of the IHAADF versus t-curve, the maximum can be exploited in some cases for thickness determination. This was shown by Pfaff et al. ~2011! who studied thin carbon and polymer films with rather homogeneous thickness. In this case, the primary electron energy for contrast reversal was deter-
mined, which corresponds to the maximum of the IHAADF versus t-curve. Either MC simulations or a suitable empirical equation, which relates the thickness at maximum IHAADF and E0 , can then be used to determine the sample thickness. Another consequence of the maximum of the IHAADF versus t-curves is an ambiguity regarding the evaluated thicknesses if preinformation on the relevant sample thickness range is not available. Information on the relevant sample thickness range is usually available if TEM samples
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are prepared by FIB milling. Without preinformation, the ambiguity can be resolved if measurements are performed for at least two different E0 values. This procedure was illustrated in the previous section for GaN and W. For evaluation of the precision of the technique the errors for IHAADF and IHAADF,s have to be considered. The experimental HAADF STEM intensities are affected by the statistical error and thickness variations because of averaging over a sample region where the latter is not an intrinsic error of the technique. The statistical error can be estimated by the M N/N criterion ~N: number of electrons in the HAADF range!. The used pixel dwell time of 30 ms and beam current of 0.62 nA give a dose of 10 6 electrons/ pixel in case of full illumination. From the simulations and average gray levels of 10,000–30,000 we estimate that 15,000– 60,000 electrons are scattered in the HAADF range. This results in a statistical error ⱕ1% and, of course, also affects the determination of IB and IW required for normalization. Gaussian error propagation yields in unfavorable cases statistical errors up to 5% for IHAADF . DIHAADF leads to an error for the sample thickness Dt, which is determined by the slope S ⫽ DI/Dt of the IHAADF,s versus t-curve. For the examples in the preceding section, it varies from 0.001 nm⫺1 ~Figs. 4b and 4c at E0 ⫽ 30 keV around t ⫽ 100 nm! to 0.01 nm⫺1 ~Fig. 12b, W at E0 ⫽ 30 keV around t ⫽ 30 nm!. The slope determines the error for the evaluated sample thickness Dt of 1–3%. Absolute values can be inferred from the thickness determination of the plane parallel W sample in the previous section. These values are significantly affected by real thickness variations of the analyzed sample region. We note that the sensitivity of low-energy HAADF STEM is high for FHBC at 15 keV with respect to small thickness changes. In Figure 4a, the error for the nominal thickness of a multiply folded film could be easily detected. Evaluation of the precision of the technique also requires the consideration of possible errors in the MC simulations. The statistical error, given by the M N/N criterion, can be neglected if the simulations are performed for a sufficiently large number of electrons. Our simulations are performed with 10 6 incident electrons, which yields N values between 10 5 and 5 ⫻ 10 5 electrons in the HAADF angular range and a statistical error ,0.3%. The measured working distance and radii of the HAADF segment have to be considered as sources for systematical errors. They determine the integration limits for evaluation of the MC simulations, as illustrated by vertical lines in Figure 2. The influence of these parameters depends on the shape of the scattering angle distribution, which is determined by the sample composition, thickness, and primary electron energy. The sum of systematical and statistical error for IHAADF,s is estimated to be below 0.5%. Larger errors can result if inadequate scattering crosssections are chosen for the MC simulations, which need to be carefully selected according to E0 and Z. SR-CSs describe the HAADF STEM intensity of low-Z materials like FHBC for E0 ⫽ 15 and 30 keV and for carbon at 30 keV. M-CSbased MC simulations yield good agreement between IHAADF
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and IHAADF,s at 30 keV for materials with intermediate and high atomic numbers, like Z ⫽ 14 ~silicon!, Z ⫽ 19 ~GaN!, and Z ⫽ 74 ~tungsten!. However, the measured intensities are consistently lower than IHAADF,s at 15 keV for carbon, silicon, and GaN although the shape of the curve and the position of maximum IHAADF resemble M-CS-based simulations for Si and GaN. We note that the discrepancy between IHAADF and IHAADF,s increases with decreasing E0 , which was observed from additional measurements at E0 ⫽ 20 and 25 keV ~not presented here!. We give two possible explanations for this discrepancy. On the one hand, the scattering cross-sections might be not applicable for the given Z and E0 values. Since the effect appears for SR-CSs ~carbon! as well as for M-CSs ~silicon and GaN!, this would indicate that neither cross-section model is suited to adequately describe IHAADF at 15 keV for materials with low and intermediate atomic numbers. On the other hand, the effect could result from yet unaddressed errors in the description of detector-related effects for which MC calculations need to be corrected. Although regarded as constant, the efficiency « and backscattering coefficient h of the detector could be energy dependent leading to exaggerated IHAADF,s at 15 keV. This hypothesis was tested by widely varying both parameters. However, the resulting effect on the IHAADF,s versus t-curve was too small to explain the deviations. The introduction of a further correction factor could formally solve the problem but lacks physical justification. Further investigations on this effect are needed. Depending on the atomic number of the sample material and E0 it is suggested that simulated and experimental HAADF STEM intensities are compared using calibration curves from reference samples with known thicknesses like the wedge samples in this work. Our results eventually shed some light on general recommendations regarding the use of SR-CSs and M-CSs because normalized IHAADF data can be used to evaluate the validity range of scattering cross-sections in MC simulations. Statements in the literature are rather vague regarding conditions for the applicability of scattering cross-sections. Browning et al. ~1994! stated that SR-CSs are sufficient for low-Z materials and high energies, while M-CSs have to be used for high-Z materials and intermediate energies ~0.1– 30 keV!. Shimizu and Ding ~1992! are more specific and suggest the use of SR-CSs in the energy range of 1–20 keV for elements lighter than aluminum ~Z ⫽ 13!. Our results in the energy range of 15–30 keV partially confirm this recommendation because SR-CSs well describe the experimental data upto Z ⫽ 6 while M-CSs are adequate for Z ⱖ 14 at 30 keV. Further experiments with suitable samples could help to define the regimes for SR-CSs and M-CSs more precisely and analyze the discrepancies between IHAADF and IHAADF,s at 15 keV. According to Czyzewski et al. ~1990!, small differences in the scattering cross-sections only cause little effects in MC simulations because of the averaging effect of the statistical scattering processes. With reference to Reimer ~1998! they state that SR-CSs and M-CSs are approximately
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equal for low-Z materials at primary energies E0 ⱖ 1 keV for scattering angles u ⱖ 108. In contrast to the suggestion of Czyzewski et al. ~1990!, IHAADF,s based on M-CSs and SRCSs differ significantly in our study for the investigated energy range. Kotera et al. ~1981! analyzed MC simulations up to 10 kV for gold and found that the SR-CSs underestimate the scattering into the HAADF angular range. This effect is confirmed in our simulations for sample thicknesses larger than the thickness at maximum IHAADF,s where the simulated intensities are generally higher if M-CSs are applied. However, in the lower thickness range before the maximum the opposite is observed, where SR-CSs give higher intensities than M-CSs. Furthermore, the intensity maximum of SR-CS-based simulations is generally higher compared to M-CS-based simulations and shifted toward lower thicknesses. Finally, different approaches exist for the calculation of M-CSs and SC-CSs in addition to the cross-sections used for the MC simulations in this work, for example, by Murata et al. ~1971!, Reimer and Lödding ~1984!, Gauvin and Drouin ~1993!, Browning et al. ~1994!, or Salvat et al. ~2005!, which need to be systematically evaluated.
HAADF STEM intensities were found at 15 keV for materials with Z-values between 6 ~carbon! and 19 ~GaN!, independent on the chosen model for the scattering cross-section. The source of this discrepancy needs further investigations. It could be related to the scattering cross-sections themselves or a yet unaddressed error in the description of detector-related effects. The accuracy of the method increases with the slope of the IHAADF versus t-curve. Accordingly, conditions should be avoided where measurements are performed close to the intensity maximum of the IHAADF versus t-curve where the slope is small. A guideline for the selection of suitable E0 values was given by calculating the thickness or mass-thickness, respectively, at maximum IHAADF as a function of Z and E0 . The main source of error can be the simulated intensities if unfavorable conditions are chosen ~scattering cross-sections, primary electron energies!. This error can be excluded if IHAADF versus t calibration curves are acquired using TEM samples with a known thickness profile like the wedge samples in this work. When choosing proper conditions, the overall accuracy of the method is better than 65% if the sample thickness is homogeneous in the analyzed region.
S UMMARY
A CKNOWLEDGMENTS
A method for thickness determination for samples with known composition is presented in this work. The procedure is based on HAADF STEM intensity measurements at primary energies E0 ⱕ 30 keV, which can be performed in an SEM equipped with a STEM detector. The normalized measured intensity IHAADF is directly compared with calculated intensities based on MC simulations. Adequate adjustment of brightness and contrast ~amplification! is required and the MC simulations need to be corrected for detectorrelated effects. The IHAADF versus t-curves are characterized by an intensity maximum at a thickness, which depends on the atomic number of the sample material and the electron energy. This can lead to ambiguous values for the sample thickness, which can be resolved by performing measurements for two different primary electron energies. The procedure was explicitly demonstrated for GaN and a W sample with a priori unknown thickness. Five samples covering a wide range of ~average! atomic numbers between 3.5 ~FHBC! to 74 ~tungsten! were investigated. Either thin films with homogeneous thickness or wedge-shaped samples with a well-defined thickness profile were studied. Experimental and calculated HAADF STEM intensities were compared at 15 and 30 keV. The MC simulations were performed with SR-CSs and M-CSs. M-CSs are a good choice at 30 keV for materials with intermediate and high atomic numbers as verified for Z ⫽ 14 ~silicon!, 19 ~GaN!, and 74 ~tungsten!. The choice of the scattering cross-section is more complex for light-weight materials. FHBC ~Z ⫽ 3.5! at 15 and 30 keV as well as carbon at 30 keV can be described well by MC simulations based on SR-CSs. Discrepancies between measured and simulated
We thank M.F.G. Klein @Light Technology Institute ~LTI!, Karlsruhe Institute of Technology# for fabrication of the FHBC films, Dr. D. Cooper ~CEA-Leti, Grenoble! for providing the SiGe sample, and Dr. D. Hu and P. Ganz @Center for Functional Nanostructures ~CFN!, Karlsruhe Institute of Technology# for providing the GaN wafer. We also acknowledge project funding by the German Research Foundation ~DFG!.
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