Imaging individual atoms inside crystals with ADF-STEM

ARTICLE IN PRESS

Ultramicroscopy 96 (2003) 251–273

Imaging individual atoms inside crystals with ADF-STEM P.M. Voyles*,1, J.L. Grazul, D.A. Muller Bell Laboratories, Lucent Technologies, 700 Mountain Ave, Rm 1D-437, Murray Hill, NJ 07974-0636, USA Received 12 July 2002; accepted 8 November 2002

Abstract The quantitative imaging of individual impurity atoms in annular dark-field scanning transmission electron microscopy (ADF-STEM) requires a clear theoretical understanding of ADF-STEM lattice imaging, nearly ideal thin samples, and careful attention to image processing. We explore the theory using plane-wave multislice simulations that show the image intensity of substitutional impurities is depth-dependent due to probe channeling, but the intensity of interstitial impurities need not be. The images are only directly interpretable in thin samples. For this reason, we ( thick, with low surface roughness describe a wedge mechanical polishing technique to produce samples less than o50 A and no amorphous surface oxide. This allows us to image individual dopants as they exist within a bulk-like silicon environment. We also discuss the image analysis techniques used to extract maximum quantitative information from the images. Based on this information, we conclude that the primary nanocluster defect responsible for the electrical inactivity of Sb in Si at high concentration consists of only two atoms. r 2003 Elsevier B.V. All rights reserved. Keywords: ADF-STEM; Z-contrast; Impurity atoms; Electron channeling

1. Introduction Imaging single atoms is one of the goals of microcharacterization of materials [1]. Many of the properties of materials are controlled by the distribution and motion of low concentrations of impurity atoms. Techniques such as X-ray absorption spectroscopy or nuclear magnetic resonance can determine the average local environment of impurities in some cases. This is insufficient *Corresponding author. Tel.: +1-608-265-6740; fax:+1-608262-8353. E-mail address: [email protected] (P.M. Voyles). 1 Current address: Department of Materials Science and Engineering, University of Wisconsin, Madison, 1509 University Ave, Madison, WI 53706-1595, USA.

information, however, if there are several different possible local environments. Imaging can characterize the environment around each impurity individually, yielding the maximum possible information. Single-atom imaging has also taken on direct technological relevance. As silicon transistors continue to shrink in size, whether or not a device works will depend on the position of just a few dopant impurity atoms [2]. Moreover, as the concentration of such impurities grows larger, they form nanoclusters, and the free carrier concentration as a function of impurity concentration saturates [3]. This may impose a fundamental limit on future generations of Si technology [4].

0304-3991/03/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0304-3991(03)00092-5

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Single atoms on surfaces have been imaged previously. The earliest such report, by Muller [5], reports imaging of atoms on a metal surface using a field-ion microscope. Imaging heavy atoms on an amorphous supporting film was one of the first applications of annular dark-field scanning transmission electron microscopy (ADF-STEM) by Crewe [6]. Similar measurements were later carefully analyzed by Treacy and Rice [7] and extended to atoms on a crystalline support by Nellist and Pennycook [8]. Atoms on a surface, however, behave very differently from atoms in the bulk. We have recently demonstrated ADF-STEM images of Si that show quantifiable contrast from a single substitutional Sb atom in the Si column [9]. We believe this is the first time single atoms have been imaged in the bulk. Zakharov et al. [10] and Takai et al. [11] suggested they could detect individual Au and Sb atoms, respectively, in Si using ‘‘manybeam dark-field lattice imaging’’. Zakharov later showed that this method failed to detect single atoms in the simple test case of Au atoms dispersed on an Si surface [12], and suggested that the method was instead measuring charged point defects [13]. With hindsight, it seems likely these are the extended, rod-like 311 defects identified by Eaglesham et al. [14] and Takeda [15]. Endoh et al. [16] reported detection of single Si atoms in an Al1.2 wt% Si alloy using Si L-edge energy-filtered imaging, but, as pointed out by several authors [17–20], Endoh et al. have neglected to consider the preservation of elastic contrast in energyfiltered images. Wang [21] has shown images of a sample without isolated impurities that have features arising from this effect that look like those in Endoh’s images. We have proposed three quantitative tests of whether or not a set of images shows contrast from single atoms [9]: (1) The images should show a number of atoms consistent with an independent measurement of the concentration. (2) The intensities of the single atoms should be consistent with a model of their spatial distribution and the relevant contrast mechanism. (3) There should be a null test; i.e., a region of the sample without impurities imaged under the same conditions (preferably at the same time) should not show

the features identified as single atoms. The previous measurements pass at most test (1) (which can be achieved simply by changing the detection threshold in the image analysis), but do not pass all three tests. Our measurements pass all three of these tests. A successful null test, test (3), is demonstrated in Fig. 1, a /1 1 0S orientation ADF-STEM image of Si doped with Sb by low-temperature MBE. Fig. 1 shows the interface region between the undoped Si substrate on the right and the Sbdoped Si on the left. In the doped region, some of the atomic columns are much brighter than the surrounding columns. These columns contain usually one, and occasionally more than one, Sb atom. In the substrate region, all of the atomic columns are similar in intensity. This shows in one image, which means there can be no difference in acquisition conditions, detector gain, or image processing, that we see features associated with single atoms where the atoms should be, and we do not see such features where the atoms should not be. Tests (1) and (2) have been discussed elsewhere [9]. The area of the sample shown in Fig. 1 has a thickness gradient from left to right, as discussed in the sample preparation section, which means that Sb atoms in the thinner part of the sample on the left have higher visibility that Sb atoms near the interface. Even near the interface, the Sb atoms have a contrast of B15%, well in excess of B5% required to distinguish them. In this paper, we first review the theory of ADFSTEM lattice imaging and its implications for imaging single atoms. Second, we present in detail the sample preparation protocol we have developed to produce the very thin samples necessary to image single atoms. Third, we discuss the image analysis techniques we employed to find the positions of the single atoms and avoid problems created by scan noise. Fourth and last we discuss our results on electrically deactivating Sb nanoclusters in Si.

2. Theory Imaging substitutional impurities provides an experimental framework in which to explore our

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Fig. 1. (a) ADF-STEM image of Sb-doped Si grown on an undoped [0 0 1] Si surface and viewed along the /1 1 0S direction. The right-hand side of the image is the undoped substrate Si, in which the atomic columns are all approximately the same intensity. The left-hand side of the image is doped region, in which there are some atomic columns that are much brighter than the surrounding columns. Most of these columns contain only one Sb atom; a few contain more than one. The position of the interface between doped and undoped material is indicated by the vertical line. (b) A line profile through the image in (a) at the position indicated by the horizontal box, also showing uniform lattice contrast on the right, and occasional high intensity columns on the left. The overall increase in intensity from left to right is due to the secondary sample thickness wedge directed inwards toward the glue line.

basic understanding of lattice imaging in ADFSTEM. We have explored these issues using planewave multislice image simulations [22], following the lead of Loane et al. [23], Hillyard and Silcox [24], and Vanfleet et al. [1]. In this view, the contribution to the ADFSTEM image from a single atom is simply proportional to the electron probe intensity at

that atom, JðrÞ: The constant of proportionality is the high-angle scattering cross section, which scales as the atomic number Z1:7 [22]. The phase of the electron wave, which is what matters in conventional high-resolution TEM, is not important. Restated mathematically, the differential contribution to the ADF-STEM image at probe position rp from one atom at lateral position ra

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and depth z; denoted dIðra; ; zÞ is simply 1:7

dIðrp ; zÞ ¼ Z Jðra  rp ; zÞ:

ð1Þ

We can then define a specimen transmission function TðrÞ as a set of Dirac d functions weighted by Z 1:7 ; one for each of the N atoms in the sample. Tðr; zÞ ¼

N X

Zi1:7 dðr  ri ; z  zi Þ:

ð2Þ

i¼1

The differential contribution to the image from the layer of atoms at a depth z is then dI ðrp ; zÞ ¼ Tðr; zÞ#Jðrp ; zÞ; ð3Þ dz where # denotes convolution. The final image intensity IðrÞ for a sample of thickness t is then Z t dI Iðrp Þ ¼ ðrp ; zÞ dz 0 dz Z t Tðr; zÞ#Jðrp ; zÞ dz: ð4Þ ¼ 0

The initial probe wave function Jðr; 0Þ is readily calculable as the Fourier transform of the objective lens aberration function, usually denoted wðkÞ: If the probe stayed in this form, it would be easy to calculate an image from Eq. (4), but that is not what happens. Instead, in a zone-axis oriented crystalline specimen, the probe channels very strongly along the atomic columns [25]. Fig. 2 shows a plane-wave multislice simulation of this phenomena. The initial probe produced by our STEM (200 kV, Cs ¼ 1:0 mm, 10 mrad aper( defocus) is placed on one side of an Si ture, 450.0 A /1 1 0S dumbbell, and we follow the probe intensity as it propagates along the atomic column. As shown in Fig. 2(b), the probe quickly becomes much more intense directly on the atomic column, ( reaching a maximum intensity around z ¼ 100 A. The probe then dechannels somewhat, spreading intensity away from the atomic column, so that the ( on-column intensity is a minimum at z ¼ 220 A. The probe then rechannels, but some intensity has spread to the adjacent atomic column. Similar calculations with similar results have been performed for Si /1 0 0S and /1 1 1S and InP /1 0 0S [23,26,27]. This phenomena can also be understood from a Bloch wave perspective, which

has been shown to agree with the plane-wave multislice methods used here [28,29]. In that picture, the on-column intensity is primarily in the symmetric Bloch states which are localized on the two atomic columns. These states act like weakly coupled oscillators: the probe intensity sloshes from one to the other, mediated by the antisymmetric states [28,29], which have greater intensity between the columns. Fig. 3 compares JðzÞ directly on the two atomic columns forming the Si /1 1 0S dumbbell to dI=dz computed by integrating over the dark-field detector at each slice. According to Eq. (1), these should be proportional to one another, and they are, within the uncertainty in the simulation, which we estimate to be 4% [30]. If we compare dI=dz to JðzÞ only of the atomic column under the beam, the two quantities are no longer proportional for ( due to contribution from the portion of z > 200 A, the probe that has spread onto the adjacent atomic column. This is consistent with Eq. (4) and the Bloch wave calculations of Findlay et al. [29] and Allen et al. [28], but contradicts the conclusions of Rafferty et al. [31] who find that the image contrast is insensitive to the effects of the probe spreading to neighboring columns. So what does the evolution of JðzÞ imply for imaging of substitutional impurities? A heavy impurity will sample the Jðr; zÞ shaped by the rest of the lattice en route to the impurity position. As discussed by Vanfleet et al. [1], this means the final image intensity of an impurity depends on its depth. For samples of thickness less than the first channeling intensity maximum, this means an impurity at the bottom of the sample will have a higher image intensity than an impurity at the top. Therefore, if the sample is thinner than the first channeling maximum, and the likelihood of having more than one impurity in a column is small (low impurity concentration and a thin sample), we might be able to determine the depth of an impurity in the column from its image intensity. If it is likely that there are two or more impurities in a column, however, not even the occupancy of the column can be determined from the intensity, since two impurities near the top of the sample can have a smaller intensity than one at the bottom. (This is the fundamental reason that

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Si

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Fig. 2. (a) Gray-scale map of the probe intensity propagating down an Si /1 1 0S atomic column. x is position across the dumbbell, z ( The probe starts exactly on the left-hand side of the Si is position along the column. The Si atomic columns are at x ¼ 0 and 1:35 A. ( defocus. (b) Surface plots of probe dumbbell at x ¼ 0: Probe conditions are 200 kV, Cs ¼ 1:0 mm, 10 mrad aperture, and 450.0 A intensity versus x and y at the indicated z:

the single- and doubly-occupied column intensity distributions in Ref. [9] overlap.) If the sample is thicker than the first channeling maximum, then intensity as a function of depth is no longer single-valued and no information about the individual column occupancies or depth of impurities can be obtained from the intensity. However, if the information we seek is only the average impurity concentration (which is one of the relevant quantities for integrated circuits, for

example) the channeling contribution averages out for thin samples [1]. At even larger thickness, we see from Fig. 3 that the probe intensity becomes greater on the neighboring atomic column than the column under that beam. The image of an impurity near the bottom of such a sample will therefore appear in the wrong atomic column! We could still determine the average dopant concentration from such an image, but the kind of atomic clustering study

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256 1.6

column under beam adjacent column

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( of both the atomic columns that form the Si /1 1 0S dumbbell for a probe that starts on Fig. 3. (a) The probe intensity within 0.03 A one side of the dumbbell. This is a one-dimensional cut through Fig. 2(a). (b) dI=dz computed by integrating over the ADF detector at each slice, compared to JðzÞ; the total probe intensity on the two atomic columns that form the Si /1 1 0S dumbbell. The two ( 1  JðzÞ: We estimate a 4% uncertainty in dI=dz and JðzÞ [22], indicated by the quantities are proportional, with dI=dz ¼ 0:2970:01 A shaded band.

reported here would be impossible. These concerns emphasize the continued necessity of using thin ( thick) samples, even (generally less than 100 A when very high spatial resolution imaging, such as with a spherical-aberration corrected STEM, is available. Applying the same ideas to imaging interstitial impurities can lead to the opposite conclusion. In some structures and orientations, such as Si /1 1 0S, an interstitial impurity may sit between

the projection of the atomic columns. (This is not always the case. For example, in Si along /1 0 0S an interstitial sits in an atomic column projection.) If the impurity is off-column, it samples a part of the probe wave function that is not enhanced by channeling, and may even be depleted. Fig. 4 shows what happens to a probe placed exactly between the two atomic columns in an Si /1 1 0S dumbbell. For parameters corresponding to our microscope, the probe is quickly channeled

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probe Si

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midpoint left column right column

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Fig. 4. (a) Gray-scale map of the probe intensity down an Si /1 1 0S dumbbell when the probe starts exactly between the two atomic ( The probe parameters are the same as in Fig. 1. (b) The probe intensity within 0.03 A ( of each atomic column, columns at x ¼ 0:68 A. and the initial x and y position of the probe as a function of z: Compare this to Fig. 3(a). As in Fig. 2, the probe is strongly channeled ( onto the atomic columns, leaving almost no intensity below the initial probe position at z ¼ 145 and 430 A.

onto the two adjacent columns, leaving little intensity at the initial probe positions between ( This effect is reduced the columns for z > 40 A. with a smaller probe, and significantly less evident at the actual Si interstitial position, which is farther away from any of the /1 1 0S atomic columns, but the general principle remains: offcolumn impurities will not see enhanced contrast due to channeling. At best the contrast will be

depth independent. At worst, off-column impurities near the bottom of samples that are not very thin may be invisible. These predictions are born out by STEM image simulations on one of the candidate Sb dopant clusters discussed in the following sections, DP2. DP2 consists of two-Sb atoms, one of which is ( away from the nearly substitutional, at 0.17 A atomic column in projection, the other of which is

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1.6

On-column Sb Off-column Sb

ISb / ISi

1.4

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Sb depth (Å)

Fig. 5. Simulated visibility (ISb =ISi ) for DP2, one of the possible electrically deactivating defect clusters. In this two-Sb cluster, one Sb, labeled ‘‘on-column’’ here, is almost precisely substitutional. The other, labeled ‘‘off-column’’ here, is displaced from ( The on-column Sb behaves like a the atomic column by 1.10 A. substitutional impurity, the visibility of which increases with Sb depth in the sample. The off-column Sb behaves like an interstitial impurity, the visibility of which is lower and independent of depth in the sample. The lines are linear fits to the simulations.

( off-column by an Si bond reconpulled 1.10 A struction (see Fig. 9). This is closer to the column ( from a than a true interstitial, which is 2.03 A column in projection. Fig. 5 shows the simulated visibility ISb =ISi for the on-column and off-column Sb atoms in a DP2 as a function of the depth of the ( thick sample. The ondefect cluster in a 45 A column Sb shows increased visibility with depth predicted for a substitutional impurity. The offcolumn Sb visibility is nearly independent of depth, and lower than that of the on-column Sb visibility. This suggests if we want to image interstitial impurities, we should either work along a zone axis that puts interstitials on-column, or, failing that, avoid high-symmetry zone axes and the strong attendant probe channeling entirely.

3. Sample preparation We start with samples of Si doped with Sb by low-temperature MBE [32]. Samples were grown

at 250 C on Si /1 0 0S to a thickness of 50 nm. The Sb concentration was 9.35  1020 cm3, measured by Rutherford backscattering spectroscopy, and the carrier concentration was 6.5  1020 cm3, measured by Hall effect. This indicates that 30% of the Sb atoms in the samples are electrically inactive. We prepared /1 1 0S oriented cross sections of the as-grown samples by wedge mechanical polishing. The method is discussed in more detail in Appendix A, and a summarized check list is provided in Appendix B. This is similar to the tripod polishing method developed by Benedict et al. [33] and Klepeis et al. [34], but with a few key differences. They were interested primarily in failure analysis of microelectronic devices, so they sought to maximize the electron transparent region of the sample and therefore the likelihood of thinning the failed device. As a result, they use a very shallow wedge angle (as low as 0.2 ) and position the material of interest perpendicular to the wedge direction (vector from thick to thin) and the direction of polishing. This orientation makes the entire front edge of the sample the area of interest, and can result in 1–2 mm of useful electron transparent sample across the front edge, with a typical thickness of 80–100 nm. However, because of the very shallow wedge angle, attempts to make the sample much thinner than this result in at best extensive curling of the area of interest, making it very difficult to properly orient the area of interest on the zone axis. At worst, the area of interest cracks or shatters. We are primarily interested in more basic science or process development, which is more likely to involve blanket films than full devices. This allows us to make two important changes to the tripod protocol. First, we orient the sample such that the area of interest runs parallel to the wedge direction and the direction of polishing. This means the area of interest will change thickness along with the rest of the sample along the wedge. Second, we increase the wedge angle to 2 . This renders the electron transparent area of the sample much smaller than the original tripod method, but for atomic-resolution characterization we only need a few hundred nm2 at most, and for a blanket film it does not matter where the thin

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area falls. The steeper wedge angle has the advantage of greatly improved mechanical stability both during polishing and subsequent handling, which allows us to mechanically polish the sample significantly thinner than is possible with a shallow wedge. For mechanical polishing we use an Allied TechPrep polishing wheel with the Multiprep sample holder assembly. Instead of using a diamond slurry or paste, we use diamond impregnated plastic lapping films as the polishing surface. These films provide a flatter, more uniform polishing surface and most importantly allow us to clean the polishing surface during polishing. This removes from the wheel the bits of sample that are polished off, preventing them from hitting the sample again and creating scratches. The use of conditioned or distilled water throughout greatly improves the cleanliness and reproducibility of the polishing. The Multiprep attachment allows us to polish with only the sample touching the polishing surface. This is in contrast to other tripod polishing jigs or lapping fixtures in which some part of the fixture also touches the polishing surface. The Multiprep also allows us to keep a constant, adjustable load on the sample, and maintain orientation much more accurately than we could achieve by hand. It also has a built-in micrometer which is useful for measuring the amount of sample polished away. We have made a few additional modifications to the Allied system, discussed in Appendix A, and shown in Fig. 6. There are two additional steps which let us achieve the very thin and very clean samples necessary to image single atoms. The first is a final surface polish on both sides of the sample with 0.02 mm colloidal silica on a felt-covered platen. The silica spheres are in a basic buffer solution to prevent them from agglomerating, so for Si this final step of the polish is as much chemical as mechanical, leaving an exceptionally flat surface. However, even for samples of other materials we have investigated, the region of the sample next to the glue line preferentially thins, which is ideal. We speculate that this may be due to the difference in hardness between the sample and the glue, or swirling of the silica along the glue line/sample

259

Fig. 6. Allied High Tech Techprep polishing wheel and Multiprep sample holding assembly. We have added the sponge to keep the lapping film clean during polishing and the fiberoptic light.

interface. This gives us a secondary wedge pointing in toward the glue line, perpendicular to the primary wedge. Fig. 7 is an optical micrograph of a sample which shows this effect: the interference fringes caused by changes in thickness bow in toward the glue line, indicating that the glue line is thinner than the surrounding material. If the glue line thins more slowly than the surrounding material, the sample looks like Fig. 8. How to avoid this is discussed in Appendix A. The thinnest areas of the sample are therefore not right at the edge of the primary wedge, but a few microns up the glue line. They are very closely connected to significantly thicker portions of the substrate and share the same orientation. We can therefore orient the sample on a thick portion of material, where the Kikuchi lines are clearly visible, then

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10 µm Fig. 7. An optical micrograph of an ideal polished Si sample still on the optical flat. Note the interference fringes, which bend in toward the glue line, indicating that the glue line is thinner than the surrounding sample.

10 µm Fig. 8. A polished sample with a swollen glue line. Notice how the interference fringes bow the opposite direction of those in Fig. 7, indicating that the glue line is thicker than the surrounding material.

move to the very thin portion with very little reorientation required. The second step that leaves us with these very clean very flat samples is a brief hydrofluoric (HF) acid dip to strip off the surface SiO2 layer immediately before the sample is loaded in the microscope. We speculate that the colloidal silica polish leaves behind a damaged amorphous Si surface layer. This layer oxidizes quickly, easily reaching maximum thickness overnight. (In our

experience, amorphous Si films have thicker native oxides than c-Si.) Removing this layer with HF leaves us with a mostly H-terminated surface which develops only a sub-monolayer coverage of oxide in the approximately 10 min it takes us to get the sample into the microscope. Areas with no oxide coverage are undisturbed crystal with no amorphous surface layer. Patches of native oxide show up as a mottling in the ADF image on a 1–2 nm length scale and are usually clearly visible after a few hours of air exposure. Re-dipping a sample that has had a native oxide reform is not as effective as the first acid dip after polishing. The thinnest parts of the samples prepared by this method are clean crystal with no amorphous ( and occasionally as layer, routinely less than 50 A, ( They have less than 10% thickness thin as 15 A. variation across a field of view of 10  10 nm, and have estimated roughness over that field of less ( rms. Our sample yield following the than 1 A protocol given in the appendices is approximately 80%.

4. Image analysis The scientific problem we sought to address with single-atom imaging was determining the structure of the primary electrically deactivating nanocluster

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of n-type dopants in Si. Two classes of defects have been proposed: structures involving two, three, or four dopants and a vacancy [35,36], and donor pair structures, which contain two dopants and no vacancy [37]. The vacancy structures are denoted Dn V ; where D is the donor species, n ¼ 124 is the number of donors, and V is the vacancy. There are two donor pair structures, one in which the donors are second nearest neighbors in a /1 1 0S direction, denoted DP2, and one in which the donors are fourth nearest neighbors in a /1 1 0S direction, denoted DP4. Ball and stick models of a /1 0 0S projection of each of these structures are shown in Fig. 9. Ab initio total energy calculations show that the D3 V and D4 V structures are exothermic [38], so it has been

DP2

DP4

Sb2V

Sb3V

Sb4V Si Sb

Fig. 9. Ball and stick models of the Sb2V, Sb3V, Sb4V, DP2 and DP4 electrically deactivating nanoclusters, viewed in /1 0 0S projection. Structural models are taken from Chadi et al. [37].

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widely believed that these are the primary deactivating defects. We wanted to identify from the images collections of Sb-containing columns consistent with various projections of the proposed nanocluster defects. Finding these clusters meant first identifying, in a statistically rigorous way, the position and intensities of the Sb-containing columns with respect to the rest of the Si lattice. The starting point for image analysis is an image like Fig. 10(a). (This is part of ‘‘Image A’’ from Ref. [9].) This shows a region of Sb-doped Si along a /1 1 0S zone axis, imaged in a JEOL 2010F STEM (200 kV, Cs ¼ 1:0 mm, 10 mrad probeforming aperture, 50 mrad ADF detector inner angle). The Si lattice is visible, with some atomic columns appearing much brighter than the surrounding columns. Most of these columns (B85%) contain one Sb atom; a few (B15%) contain two. The first step is to strip the Si lattice out of the image. This is necessary so that the particle-counting software we will use later does not have to contend with a constantly varying background intensity. Fourier filtering, while familiar to many microscopists, is not well suited to this task. Not only will it introduce artifacts from aliasing, but the objects we wish to identify are separated by multiples of the lattice periodicity. Instead of Fourier methods, we applied singular value decomposition (SVD) analysis (see Appendix C). SVD is similar to the eigenvalue decomposition of a matrix, but is more numerically stable and applies to rectangular and singular matrices (for a mathematical treatment, see Ref. [39]). The image components with the largest singular values contain most of the variance in image intensity, which in this case is the atomic lattice. The smaller singular value image components tend to contain one or two bright, localized features each, which in this case are the Sbcontaining columns. Fig. 10(b) shows the image reconstructed from the portion of the SVD containing the lattice, which is singular values 1–8, in order from largest to smallest. The singular value 0 image component contains most of the smoothly varying background. Fig. 10(c) shows the image reconstructed from the portion of the

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SVD containing the Sb atoms, which is singular values 9–480. Singular values 481 and higher contain mostly pixel-by-pixel noise. The second step is to identify the positions of the Sb atoms in the SVD image. For this, we use a piece of software called Source Extractor [40], which is commonly used by optical astronomers to identify galaxies in crowded star fields [41]. Source Extractor identifies objects with reference to a measure of the local background variability, s: This is an advantage over particle-counting software more commonly used in electron microscopy such as NIH Image which uses a global intensity threshold. Changes in local lattice background might arise from some residual surface roughness, thickness variations, or small patches of oxide. Source Extractor identifies as objects collections of pixels of a certain minimum size, all of which are above a certain intensity, measured in units of the per pixel standard deviation of the local background, s: It also applies a ‘‘deblending’’ algorithm to separate closely spaced objects which are not fully resolved [40]. We identified objects as collections of six pixels ( that are all above a (for Fig. 10 one pixel is 0.25 A), 1:5s per pixel threshold from a 64  64 pixel local background area. Source Extractor was also used to apply a 2.0 pixel full-width at half-maximum 3.0  3.0 pixel Gaussian convolution filter before object detection. This places the per object chance of a false detection due to a random fluctuation in the background at the 9s level. This is small enough to be insensitive to changes in the detection threshold. The threshold was therefore set by processing an image containing both doped and undoped regions, and selecting the minimum threshold that gave no detections in the undoped region. This gives an Sb detection efficiency of B100%. A higher threshold simply reduces the detection efficiency. We estimate that the residual detection error between 0 and 1 Sb atoms with a 1:5s per pixel threshold is 3%. Fig. 11(b) shows the detected objects from Fig. 10(c). A few

(a)

(b)

(c)

10 Å

Fig. 10. (a) Experimental image of Sb-doped Si, unprocessed except for some smoothing to reduce noise. (b) Lattice portion of the image from the SVD, also smoothed. (c) Sb portion of the image from the SVD, also smoothed.

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(a)

(b)

10 Å Fig. 11. (a) Lattice portion of the image in Fig. 10 with dots indicating the column positions and lines indicating the nearestneighbor connections between columns. (b) Sb portion of the image in Fig. 10 with circles indicating the Sb atom positions and lines indicating nearest-neighbor connections between them.

detected objects (B2% in Fig. 11(b)) were closer together than the width of the point-spread ( These objects function of our STEM, B1.6 A. were combined with one another. We initially tried to find Sb nanoclusters simply by analyzing the Sb atom positions provided by

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Source Extractor, but this failed for two reasons. First, the position of the very bright features may be shifted a few pixels in the smoothed SVD reconstruction compared to the original image. Second, and more of a problem, our STEM has some residual scan noise. (Replacing the DIGISCAN cables supplied by Gatan with properly shielded cables has subsequently removed much of the noise present in the images shown here.) Unlike in a TEM, where instability results in reduction of contrast and aberrations like a pincushion distortion result in a global and predictable deflection of the lattice fringes, scan noise in a STEM results in local deflection of the lattice fringes without a predictable global pattern. This case been seen along the left-hand side of Fig. 10(b), where the almost vertical /2 2 0S fringe is not straight. This caused an analysis based on intercolumn distances and angles to yield inconsistent results across the image field. The column positions are not of fundamental interest at any rate, at least not at the resolution achievable in our non-Cs corrected STEM. Instead, what we want to know is the cluster topology: are the Sb atoms nearest neighbors, or next-nearest neighbors? To obtain this information, we first extracted all the column positions from the lattice portion of the SVD (Fig. 10(b)) using Source Extractor (0:75s threshold over 5 pixels, 1.5 pixel FWHM Gaussian filter). Then we mapped those positions onto a two-dimensional, four-connected lattice formed by the {1 1 1} lattice planes. The result is shown in Fig. 11(a). Each Sb atom was mapped onto the nearest lattice column position, which also determined its connectivity to the surrounding Sb columns, indicated by the lines Fig. 11(b). Unfortunately, not all the defect structures are distinguishable in /1 1 0S projection at the resolution of our STEM. In particular, the DP2 and Sb2V structures are both next-nearest-neighbor Sb atoms, the difference being the presence or absence of an invisible vacancy. These structures are indistinguishable in all projections. Moreover, projections exist of DP2 and Sb2V in which both Sb atoms lie in the same atomic column. Sb4V looks the same in every projection. Sb3V looks like Sb4V in two projections, and like DP2/Sb2V in the

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other. The six unique images and their possible structural assignments are shown in Fig. 12. Some structures in some projections have columns containing two-Sb atoms, but it is not possible to use the experimental intensities to determine the column occupancy reliably for these samples [9]. The fraction of Sb-containing columns that are involved in each one of these structure images and the implications for the primary deactivating defect are discussed in Section 5. There is one additional step we can take in the analysis, which rests on the observation that the

experiment 2D random occupancy

120

100

Sb columns

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80

60

40

20

0 1

2

3

≥4

cluster size

Fig. 13. The two-dimensional cluster sizes calculated for random column occupancy and from the image, which are in excellent agreement.

(a)

(b)

(c)

(d)

(e)

(f)

distribution of Sb-containing columns is statistically random in two dimensions. We can calculate, for example, number of nearest-neighbor clusters on the 2D {1 1 1} lattice as a function of the cluster size for random column occupancy at the observed total occupied column concentration from perimeter polynomials [42]. Fig. 13 shows that the experimental data match the calculation for a random distribution of Sb atoms for all cluster sizes. Since the image field shows both the growth direction (left to right in Fig. 10(a)) and a perpendicular direction, we can assume that the Sb atom distribution is very close to random in all three dimensions. Under this assumption, we constructed full 3D models of the Sb distribution in our sample which are consistent with the images. We took the x and y positions of each occupied column from the image, and populated each column with a random number of Sb atoms consistent with a binomial distribution,

10 Å

Pn;c ðsÞ ¼ Fig. 12. The six possible images generated by all the possible /1 1 0S projections of the DP2, DP4, Sb2V, Sb3V, and Sb4V defect structures. The structures that are consistent with each image are (a) isolated Sb, DP2, or Sb2V; (b) DP2 or Sb2V; (c) DP2 or Sb2V; (d) Sb3V or Sb4V; (e) DP4; and (f) DP4. Each image could also be generated by Sb’s isolated in depth.

n! cs ð1  cÞns ; s!ðn  sÞ!

ð5Þ

where s is the number of Sb atoms, n is the total number of atoms in the column, and c is the Sb concentration. c is known from RBS measurements (c ¼ 0:0187). n is proportional to the

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thickness, which we measured by comparing the ratio of the /1 1 1S to /0S beam intensities in a CBED pattern to multislice simulations. For Fig. 10, n ¼ 9 atoms. We placed each of the Sb atoms in the column at a random z; avoiding Sb– Sb nearest-neighbors, which are inconsistent with X-ray absorption data [37]. From each 3D model, we calculated the number of DP4, Sb3V, and Sb4V defects. DP2 and Sb2V defects remain indistinguishable, but we calculated the number of defects that were consistent with both structures. We then iterated over many randomly generated 3D structures, each consistent with the image data, to measure the properties of the real sample. Results of this analysis are given in Section 5.

5. Results 5.1. Limits on cluster statistics from the 2D images The percentage of columns involved in a twodimensional cluster that could be a defect is given for each defect type in Table 1. Now we consider whether any of these measurements are consistent with the observed electrically inactive fraction f ¼ 30%: What if Sb4V, the most energetically favorable structure, is the primary deactivating defect? If we assume that every triplet of occupied columns is an Sb4V defect, then f ¼ 4=3  11% ¼ 15ð3Þ%; which is inconsistent with the known electrically inactive fraction. (The 4/3 arises because one of the columns in the triplet contains two-Sb atoms.)

Table 1 Percentage of columns involved in clusters that could possibly be the indicated type Cluster type

% of columns

DP2 or Sb2V DP2, Sb2V, or Sb3V DP4 Sb3V/Sb4V

62 50 46 11

(5) (5) (4) (2)

Note: The value in parentheses is the estimated standard error, primary from counting statistics. There are 240 Sb-occupied columns in the image.

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We emphasize that this is a deliberately high upper bound on f ; since some of the triplets in the image are bound to really be a pair defect with a third Sb nearby in projection but separated in depth, or three electrically active Sb’s, all separated in depth. What about Sb3V? If we assume that all the triplets are Sb3V and all three of the orientations of Sb3V are equally likely, then a number of pairs equal to half the number of triplets are actually Sb3V defects. There are more than enough pairs for this to happen. This makes f ¼ 3=2  11% ¼ 16ð3Þ%: Again, this deliberate overestimate is inconsistent with the known value of 30%. What about DP4? Similar to Sb3V, if all the possible fourth-neighbor pairs are DP4 and all three orientations are equally likely, f ¼ 3=2  52% ¼ 78ð5Þ; which is far too high. Remember, however, that this is an overestimate of the real f ; so all we can say is that this is not inconsistent with the measured value. The situation for DP2/Sb2V is similar; various ways of accounting for differing decompositions of triplets into pairs and singles all lead to f > 50%: We can conclude, however, based solely on the two-dimensional distribution of the occupied columns, that the energetically favorable Sb3V and Sb4V defects cannot be the primary deactivating defect. Even under the most generous assumptions, there just are not enough of them in the images. Analysis of another image similar to that shown in Fig. 10 (Image ‘‘B’’ from Ref. [9]) leads to the same conclusion. 5.2. Cluster statistics from 3D Monte-Carlo analysis To determine whether any of the pair defects are consistent with the measured f ; we turn to the three-dimensional Monte-Carlo analysis results, summarized in Table 2. Any of the pair defect structures, taken one at a time, give excellent agreement with f ¼ 30%: Again, analysis of another image similar to Fig. 10 yields leads to the same conclusion. This indicates that many of the apparent pairs in the images are actually two active Sb atoms separate in depth. If we allow both types of donor pair defects simultaneously, f rises to an unacceptably high value. This could be

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Table 2 Percentage of atoms involved in clusters of the indicated types from 3D Monte-Carlo analysis Cluster type

Electrically inactive fraction

DP2/Sb2V DP4 DP2/Sb2V and DP4 Sb3V Sb4V

33 (3) 29 (3) 53 (3) 1 (3) 0 (3)a

Note: The 3% value in parentheses is the estimated standard error from counting statistics. a No Sb4V configurations were observed in any of the 60 structures, each consistent with the image data, that were randomly generated.

evidence that the primary defect is Sb2V, unless kinetic conditions favor one DP defect over the other. There are almost no Sb3V or Sb4V defects in the 3D structures; anything that looks like one in the image is a chance coincidence viewed in projection.

6. Conclusions We have imaged individual Sb atoms still bonded inside an Si crystal using atomic-resolution high-angle ADF-STEM. The necessary samples ( thick were prepared by wedge mechanical o50 A polishing. The Sb atom positions and cluster topologies were determined from these images. We conclude based on this data that the primary defect structure which causes the carrier concentration to saturate as a function of Sb concentration involves only two-Sb atoms, not three or four as had been previously believed. Probe channeling renders the intensity scattered from a single substitutional impurity depth-dependent. Under conditions of a very thin sample and low impurity concentration, this may allow the determination of the depth of the impurity in the column. With thicker samples and higher concentrations, only the average concentration can be determined. For very thick samples, coupling between columns could lead to the image of an impurity appearing in the wrong atomic column. Beyond this particular problem, the ability to acquire images which show contrast from indivi-

dual impurity atoms will allow electron microscopy to address problems that were previously inaccessible. Single-atom sensitivity should be achievable in Si for any atomic species heavier than Sb. Our experiments suggest that As atoms might also be visible in samples thinner than ( In general, any impurity in a matrix such 100 A. that Zimpurity =Zmatrix X3 should be visible, provided the sample can be made thin, smooth, and clean enough. We have thinned samples of GaN, ( using the SrTiO3, and InP to less than 100 A wedge-polishing technique described above. Combine this type of imaging with the ability to do in situ measurements at elevated temperature [43], and this technique will be a general tool for studying the distribution, diffusion, and clustering of impurities in materials.

Acknowledgements We thank H.-J.L. Gossmann for providing the Sb-doped Si samples we examined, D. J. Chadi for defect structural models, and P. H. Citrin for introducing us to the problem of deactivating nanoclusters in Si. We thank Partha Mitra for introducing us to singular value decomposition, and David Wittman for help with Source Extractor.

Appendix A. Sample preparation protocol The general outline of the procedure is as follows: First, we prepare a sandwich of two pieces of material glued ‘‘butter-side in’’, cut the sandwich to size, then polish one side of sandwich, then the other. Then we mount the polished sandwich on a grid and perform a final acid etch right before the sample goes into the microscope. A checklist is provided in Appendix B. As this is a tripod polishing technique, more helpful hints can be found in the articles by Benedict et al. [33] and Klepeis et al. [34]. A.1. Making the sandwich To make the sandwich, we first cleave two pieces approximately 3  5 mm off the main sample

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block. The two pieces are thoroughly cleaned of debris, carefully examined, then glued face to face using Allied Epoxy Bond 110. The glue layer between the sample pieces should be as thin as possible. If it is too thick, the sandwich will not polish well and has a high probability of falling apart. We find that the viscosity of the Allied 110 epoxy particular lends itself to thin glue lines; o1 mm is achieved routinely, and the best glue lines are about 100 nm thick. If you can see a glue line after polishing with the 30 mm paper, it is too thick. Also, this epoxy is mixed immediately before using, so it is always ‘‘fresh’’. We still check that the epoxy is in good condition by allowing a drop of it to polymerize on a glass slide before gluing the sandwich. If the drop does not turn a deep red then either the mix is incorrect or the resin and the hardener have become hydrated through improper storage. We clamp the sandwich together under a stereomicroscope to be sure that the two pieces are even and level, and let it sit for 5 min while the glue distributes itself. The sample is then placed on a 160 C hotplate for 1 h to polymerize. This is longer than the manufacturer suggests, but it gives a more chemically resistant glue line. The sample is then glued to a graphite block with either Apiezon black wax or Crystal Bond thermoplastic wax and cut to length with a diamond impregnated 0.0100 wire using a South Bay wire saw. The orientation of the saw-cut will define the surface of the sample, so some care should be taken to position the sample as close to the desired plane as possible. The pieces that are cut off must easily fit into 1  1.5 mm slot in a standard slot grid after polishing. Either of the waxes is easily removed using either trichloroethylene (TCE) or acetone. This avoids using dimethylformamide (DMF), which will attack the Allied epoxy in the glue line and may cause the sandwich to come apart. (When the sandwich breaks, it usually does so when we are trying to mount it on the slot grid, after all the time has been invested polishing.) After cleaning the sample with TCE, acetone, and methanol we place the sample on a fresh piece of filter paper to dry on the hot plate. In general, we use solvents in the sequence DMF, TCE, acetone, and methanol, rinsing off each solvent with the next in the sequence and never

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allowing the sample to dry until the end. In our experience, methanol or ethanol are the cleanest of the commercially available solvents. A.2. First-side polish A detailed polishing recipe, describing how much material is removed on which grit paper at what wheel speed is given in Appendix B. In general, the ‘‘rule of three’’, or the ‘‘trinity of damage’’ dictates the polishing recipe. During mechanical polishing a damage layer forms with a thickness approximately three times the size of the grit used to do the polishing (so the 30 mm diamond lapping film causes at least 90 mm of damage to the polished surface). Each subsequent film must remove the damage that has been created. For the first-side polish we attach the sample to an Allied aluminum-polishing stub using Crystal Bond. We then measure the thickness of the sample on the stub using a digital micrometer. We want the sample to be a little over 1 mm thick; the initial polish with the 30 mm diamond lapping film will bring the sample to the 1 mm point. Of course, if there are crevasses, chips, or gross misalignments we polish through them so that we have two relatively even pieces of sample to work with. The condition of the diamond lapping film during polishing is crucial. Before using each piece of film, we examine it carefully for nicks, scratches, or small regions where the diamond has been worn away. If such problems exist, particularly for film with finer than 15 mm grit, we discard that piece of film and use a fresh sheet; hitting any of these imperfections with the sample will create deep scratches. Discarding the paper is expensive, but less so than losing a crucial sample! It is also necessary to keep the film clean during polishing. Otherwise, bits of the sample that have been polished off will cause scratches. We clean the film by holding a folded piece of paper towel or sponge firmly on the polishing surface, moving to a fresh part of the towel once it has picked up enough polished Si to turn dark brown or black. Bits of the sample that break away from the thin edge are caught by the sponge/towel and removed before they can create scratches. A jig holding a

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small rotating sponge like those used for applying paint can replace the hand-held paper towel for all but the smallest grits (1.0 mm and smaller), as shown in Fig. 6. Finally, after polishing, we briefly rinse the film, then wipe it off with a rubber squeegee. This removes most of the residual polished material and extends the life of the film. For all but the two smallest grit films, we use conditioned water with a neutral pH and very low mineral content as a lubricant. Distilled water would also be ideal. We find that the lapping films remain cleaner and give us more consistent results when we use the conditioned water instead of local tap water, which tends to be hard at our location. The conditioned water also greatly reduces the risk of colloidal silica sticking to the sample after the final polish. This can occur if the silica is at the wrong pH, so using buffered water on the wheel is critical. The two smallest grit films (0.5 and 0.1 mm), require the use of diamond extender lube, because the lower surface tension with the paper seems to reduce sample cracking. For Si samples we use the water-based extender, which is green in color. Anhydrous lubricants are also available, and we have used them to successfully polish hygroscopic samples. Before and during the small-grit (0.5 and 0.1 mm) polishes we carefully inspect the sample using a compound light microscope. If the glue line is too thick, or if there are any other anomalies, we abandon the sample and start over. After the 0.1 mm polish, we clean and dry the sample, then inspect the surface with a light microscope at 1000  in dark field to see if there are any scratches or debris, especially near the glue line. If there are gross scratches then we polish further with the 0.1 mm film, or change the film, so that the scratches are not present after the next inspection. If there is debris, then we heartily clean the surface with methanol followed by distilled water. To do the silica polish, we first rinse the platen with the conditioned water while we feel the surface for debris or imperfections in the felt. Then we apply the colloidal silica and again feel the surface, this time for aggregated silica chunks. If the platen passes these two inspections the sample is slowly lowered toward the platen. For this step and the 0.1 mm film polish, the Multiprep

micrometer is not very reliable, so we rely on visual confirmation of contact between the sample and the felt. We have installed a fiber-optic light just above the sample as shown in Fig. 6 so that we can see the sample contact the surface of the felt and leave a trail in the colloidal silica as the platen turns. Once the sample has been polished, we must remove the silica that may be stuck to the surface of the sample. First, with the wheel running, we turn on the conditioned water for 2 min. Then we clean the sample with micro-organic soap under running conditioned water using a soft applicator stick. Wipe by rolling the applicator backwards, against the direction of the wipe. Never let the same area of the applicator touch the sample twice. This is followed by a rinse with distilled water and a final inspection before second-side polish. If there is silica visible on the surface under the light microscope, repeat the silica polish and subsequent cleaning to remove it. If the silica has coagulated on the sample, there may be a problem with the pH of the rinsing water. The sample is removed from the aluminum stub by heating the stub on the hot plate and lifting the sample off. We do not clean the Crystal Bond off the sample at this time. Immersing the thin sample in solvents can cause the glue line to swell, distorting the sample just in the region of interest, and removing the beneficial preferential thinning from the silica polish. Such a sample is also more likely to break due to stress from the swollen glue line. A sample with this problem is shown in Fig. 8. We will polish off any remaining Crystal Bond at the start of the second-side polish. A.3. Second-side polish For second-side polishing, we attach the sample to an Allied optical flat. We use a transparent stub so we can judge the thickness of the sample using the color of transmitted light. The right epoxy to use to attach the sample to the optical flat is the subject of some internal lab debate. In general, we require a very thin layer of glue with no trapped air bubbles. We use either more Crystal Bond or Locktite 460 superglue. Using Crystal Bond is simpler and faster, but can be less reliable. In

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particular, the Crystal Bond underlayer tends to be much thicker, and the wedge can flex more, making cracking in thin regions more likely, and even worse, rounding out the tip of the wedge. Silica particles can work in between the crystal bond and the sample, and these cannot be cleaned off. We sometimes see residual silica particles in the TEM, and this is their most likely origin. We heat a planarized flat and melt a small amount of Crystal Bond on the edge where the sample will go. The aluminum sample holder is heated next to the hot optical flat. Putting a beaker over the holders heats them up more quickly. When the Crystal Bond on the flat is very hot and runny, we remove the sample from the aluminum block with clean tweezers, flip it over and place it polished side down on the optical flat in the hot Crystal Bond. The hot flat can be moved to a stereomicroscope for final adjustments of the sample using a toothpick, including a little nudge downwards to purge air bubbles under the sample. After the flat has cooled, we start second-side polishing. To use superglue, we first remove the sample from the aluminum block using the hotplate. When the sample is cool we inspect the polished side of the sample under a stereomicroscope for debris and dust, and clean it, if required, using a clean camel hair paintbrush. We then position a clean, planarized optical flat next to the sample. The polished side of the sample is placed down on the optical flat, opposite from the region we intend to glue the sample. A small amount of superglue, which we store in a dessicator, is placed on the flat using a toothpick. We make sure that there is no debris or air bubbles in the glue. The sample is then gently placed into the glue and given a nudge into place using a torn piece of filter paper. The excess glue is wicked away by the torn paper. A beaker is placed over the flat and the glue is left to dry for a minimum of 1 h, though several hours is preferable. Putting the sample on the flat in a vacuum dessicator can speed the drying somewhat. Very hard samples like sapphire or silicon carbide must be attached to the flat with Locktite rather than wax or the samples will slide off the flat rather quickly. We polish our samples, even Si, at a wedge angle of 2 , which is much larger than the angle

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commonly used for tripod polishing. We find that a 2 wedge decreases the chance of the sample curling, makes the sample more robust to handle, and still leaves us with sufficient thin area near the glue line. This is aided by the preferential thinning near the glue line caused by the silica polish. During the second-side polish when the sample gets thinner than 60 mm, we decrease the load on the Multiprep polishing head to zero. This keeps the sample from bouncing, which prevents damage from occurring at the edge of the wedge. However, decreasing the load on the sample makes the head micrometer less reliable, so we watch the sample polish through the optical flat for added security. The fiber-optic light is very useful at this stage. (We have accidentally polished away entire samples by watching the micrometer instead of the sample!) The final stages of the second-side polish are less formulaic than the rest of the process. Once we have polished to 60 mm thickness using 5 mm paper, we remove sample in 10 mm increments using 1 mm film at 50 rpm, inspecting the condition of the wedge under the light microscope each time. For an Si sample, we are looking along the end of the wedge for the red color that indicates Si less than 10 mm thick. We are also looking for a level sample area with no gross scratches, and rainbow interference fringes along the glue line, as shown in Fig. 7. If we find all these things, we are ready to continue with a fresh 0.5 mm film at 30 rpm with the Allied green diamond extender as the lubricant. We polish the sample for 30 s while carefully watching the sample. The sample is rinsed with conditioned water and inspected. There should be fewer scratches and the interference fringes should be more defined. If the sample looks right then we proceed to a fresh 0.1 mm film at 30 rpm using the green diamond extender for 30 s. We rinse and inspect. During these final steps if at any time the sample does not pass the inspection we continue to polish with the current film before proceeding to the next film. If the damage on the sample persists, we switch to another fresh piece of film, as the 0.5 and 0.1 mm film is occasionally defective. We finish with a silica polish using the same protocol for the first-side polish.

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After rinsing and inspecting for debris we are ready to remove the polished sample from the optical flat. Whether we use the Locktite glue or Crystal Bond, we prefer to use acetone to remove the sample. We also sometimes use DMF, which is faster, but we must be careful to never leave the wedge in the DMF for longer than is necessary, since DMF will attack the glue line. Getting the thin sample from solvent to solvent, then finally out of the methanol can be a little tricky. We start by placing a piece of fresh filter paper with one edge folded up in a Petri dish. We then place the optical flat with sample attached on top of the filter paper, add the acetone or DMF, and put the dish on the hot plate. We watch for the sample to fall off onto the filter paper. When the sample falls off, we remove the flat, pipette off the first solvent, and gently replace it with the next, through the solvent sequence ending with methanol. After 1 min in the methanol, we remove the filter paper via the folded side and place it on the hot plate. When the methanol has evaporated the sample is ready to be mounted on a molybdenum slot grid. A.4. Sample mounting and acid dip We mount the sample on the grid using a small dot of MBond 610 epoxy. We pick up a grid in a clean pair of self-closing tweezers, then put the epoxy (just a dot) on the grid. We let the epoxy dry slightly so it becomes tacky, then use the grid to pick the sample from the filter paper it was resting on from the last step. We let the sample dry for an hour at room temperature, then put it on the hot plate for an hour to cure. The initial drying prevents the sample from creeping back up onto the grid. For a short sample piece, the rapid curing can move the thin part of the sample out from the slot in the grid, rendering the sample useless. The final step in the sample prep, the acid dip, is done right before the sample goes in the microscope. The silica polish which is the last step on both sides leaves behind a damage layer just like the other polishing steps. This amorphous silicon oxides quickly in air. We use a brief HF acid dip to remove the SiO2, leaving a nearly pristine crystal for viewing. The crystal will re-oxidize in air within a few hours, creating an oxide that is more difficult

to remove, so this step should be done only right before the sample goes in the microscope. If a sample is repeatedly re-dipped, it becomes rougher and more facetted, making it harder to see subsurface features. The protocol for the acid dip requires two Teflon beakers, one Pyrex beaker, distilled water, a disposable plastic pipette, and 48–51% HF acid. We first rinse all of the beakers with distilled water. Then we put 50 ml of distilled water into each of the beakers, and 0.5 ml of the HF in the first Teflon beaker. With the grid in a pair of self-closing tweezers we dip it in the 1% HF solution for 1 min, slowly swishing the sample. We then rinse the sample in the second Teflon beaker, which contains only water, for 30 s, in the Pyrex beaker for 30 s. The sample is then left in the tweezers and placed on the hotplate for 2 min. We use a little scrap of filter paper to remove the water meniscus from between the jaws of the tweezers, and then to give the sample a slight push into its box. The key to a successful dip is cleanliness. For the first time, it is best to start with new beakers and tweezers, and not use them for anything else. The tweezers must never have touched any copper, and copper grids should not be used. Cu is soluble in the acid, and precipitates out on the sample, especially at the edges.

Appendix B. Polishing recipe This section presents a summary checklist of the protocol we use to thin the Si cross-section samples from the bulk. The guiding principles of our method and some special concerns are discussed above in the Sample Preparation section and Appendix A. The primary tool we use is an Allied High Tech Techprep polishing wheel with Multiprep sample holder attachment. We also used Allied diamond-coated polishing paper and other polishing supplies. B.1. First-side polish 1. Mount the sample on an Allied aluminumpolishing block saw-cut side down using Crystal Bond thermoplastic wax.

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2. Planarize the Multiprep polishing head following the procedure in the Allied manual. 3. Thin to 1000 mm on 30 mm diamond paper at 100 rpm. 4. Thin to 700 mm on 15 mm diamond paper at 100 rpm. 5. Thin to 650 mm on 6 mm diamond paper at 75 rpm. 6. Thin to 600 mm on 3 mm diamond paper at 50 rpm. 7. Remove 10 mm with the 1 mm diamond paper at 50 rpm. 8. Inspect the sample carefully under a light microscope. 9. Remove 3–5 mm on 0.5 mm diamond paper at 30 rpm, using Allied Green Lube instead of water to lubricate the paper. 10. Remove another 1 mm on 0.1 mm diamond paper at 30 rpm, still using Green Lube. This takes between 30 s and 1 min. 11. Do a final polish on 0.02 mm blue colloidal silica on Allied Final A polishing cloth at 50 rpm: First flood the cloth with water, then turn the water off and pour on the silica. Polish for 2 min, then rise off the silica with water for 2 min. 12. Clean the sample using micro-organic soap and a soft applicator stick. 13. Heat the polishing block on a hot plate and lift off the sample. Do not remove the Crystal Bond on the surface of the sample with solvents; it will be polished away in subsequent steps. B.2. Second-side polish 1. Planarize an optical flat polishing block. 2. Mount the sample first-side down on the optical flat using Crystal Bond or Locktite 460 superglue. 3. If the sample is >700 mm thick, thin to 500 mm on 30 mm diamond paper at 100 rpm. 4. Start the wedge. A 2 wedge is one full rotation clockwise on the front micrometer. For Si samples, we use a 2 wedge. 5. Thin to 250 mm on 15 mm diamond paper at 100 rpm. 6. Thin to 150 mm on 6 mm diamond paper at 75 rpm.

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7. Thin to 60 mm on 3 mm diamond paper at 50 rpm. 8. Thin on 1 mm diamond paper until the sample is red at the edge under a light microscope, checking every 10 mm. 9. Correct the tilt with the left–right micrometer. 10. Remove 3–5 mm on a new piece of 0.5 mm diamond paper at 30 rpm, using Green Lube instead of water. 11. Polish the wedge to the region of interest using a new piece of 0.1 mm diamond paper using Green Lube. 12. Do a final polish on the 0.02 mm blue colloidal silica on Allied Final A polishing cloth at 30 rpm. First flood the cloth with water, then turn the water off and pour on the silica. Polish for 2 min, then rinse off the silica for 2 min. 13. Clean the sample using micro-organic soap and a soft applicator stick. 14. Remove the sample from the optical flat in heated acetone. This should take about 5 min. If after 5 min the sample is not free of the flat, tap the back of wedge with the tip of a paintbrush to free it. 15. Rinse the sample with hot methanol. Wash off each solvent with the next, never letting the sample dry. 16. After the methanol rinse, wash sample onto filter paper and let it dry. 17. Mount the finished sample on a clean slot grid by putting a small drop of Mbond 610 epoxy on the ring, letting it dry a little so it becomes tacky, then using the ring to pick up the sample off the filter paper. Let the epoxy dry for an hour, then cure for another hour on the hot plate. 18. Immediately before the sample goes into the microscope, remove the surface oxide with an HF acid dip.

Appendix C. Lattice image processing using singular value decompositions Singular value decompositions [39] have been used in image processing and compression [44], and lie at the heart of principal components

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analysis of large data sets, especially spectral images [45,46]. The SVD can be thought of as a generalization of an eigenvalue decomposition of a symmetric matrix to the asymmetric case (a simple introduction is given by Kalman [47]). The SVD is defined as follows: if the original image X has m  n pixels, it can be decomposed as X ¼ USV T ;

ðC:1Þ

where U and V are unitary matrices with m  m and n  n elements, respectively, and S is a diagonal matrix of the same dimensions as X : The diagonal entries S (i.e. Sii ¼ si ) can be arranged to be nonnegative and in order of decreasing magnitude. The nonzero si are the singular values of X : The columns of U and V are the left and right singular vectors of X : If X were a symmetric matrix then si would be the eigenvalues, U and V would be the same, and their columns would be the eigenvectors. In MATLAB, the SVD can be performed as ½u; s; v ¼ svdðxÞ: The ranking of the singular values is important as this reflects the fraction of the image variance that is captured by that singular mode. A reconstructed image X 1 using the first k1 modes can be obtained from the transform X 1 ¼ US1 V T ;

ðC:2Þ

where U and V now contain only the first k1 columns and S1 ¼ diagðs1 ; s2 ; y; sk1 Þ: In MATLAB the reconstruction would be performed as recon ¼ uð:; 1 : k1Þ sð1 : k1; 1 : k1Þ vð:; 1 : k1Þ0 ; The first singular value (and vector) captures the image background. This is the term s1 u1 vT1 where u and v are column vectors of U; V : Each singular mode contributes a similar outer product, which has the dimensions of X : Two singular modes are needed to describe each lattice periodicity (or periodic scan noise!), unless they are exactly vertical or horizontal in which case only one singular value is needed. In a large field of view, the lattice fringes (which are periodic across the image) should be the next few components. In our images for instance, the two sets of {1 1 1} fringes, and the {2 2 0} spacing accounted for the next six modes. The dopant atoms are randomly distributed so very little image variance is captured by a single dopant atom and a few hundred modes are

required to describe them. The remaining modes contain the image noise, and the precise cutoff between signal and noise is a matter of inspection of the image formed by the weighted outer product si ui vTi : The SVD works well in removing the lattice when there is no other structure in the image beyond the dopant atoms and the lattice.

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