The invention relates to a method of examining a sample in a charged-particle microscope of a scanning transmission type, comprising the following steps:
The invention also relates to a charged-particle microscope in which such a method can be performed.
As used throughout this text, the ensuing terms should be interpreted so as to be consistent with the following explanation:
Charged-particle microscopy is a well-known and increasingly important technique for imaging microscopic objects, particularly in the form of electron microscopy. Historically, the basic genus of electron microscope has undergone evolution into a number of well-known apparatus species, such as the Transmission Electron Microscope (TEM), Scanning Electron Microscope (SEM), and Scanning Transmission Electron Microscope (STEM), and also into various sub-species, such as so-called “dual-beam” tools (e.g. a FIB-SEM), which additionally employ a “machining” Focused Ion Beam (FIB), allowing supportive activities such as ion-beam milling or Ion-Beam-Induced Deposition (IBID), for example. In a TEM, the electron beam used to irradiate a sample will generally be of significantly higher energy than in the case of a SEM (e.g. 300 keV vs. 10 keV), so as to allow its constituent electrons to penetrate the full depth of the sample; for related reasons, a sample investigated in a TEM will also generally need to be thinner than one investigated in a SEM. In traditional electron microscopes, the imaging beam is “on” for an extended period of time during a given imaging capture; however, electron microscopes are also available in which imaging occurs on the basis of a relatively short “flash” or “burst” of electrons, such an approach being of possible benefit when attempting to image moving samples or radiation-sensitive specimens, for example. More information on some of the topics elucidated here can, for example, be gleaned from the following Wikipedia links:
A (S)TEM is a relatively versatile tool, and can be used in a variety of modes to investigate a sample. Apart from “conventional” TEM imaging, there are also specialized STEM techniques such as BF (Bright Field), ADF (Annular Dark Field) and HAADF (High-Angle ADF) imaging, e.g. as alluded to in the following Wikipedia links:
It is an object of the invention to address these issues. In particular, it is an object of the invention to provide a radically new method of investigating a sample with an STCPM. Moreover, it is an object of the invention that this method should make more efficient use of available resources, and provide results not currently attainable with prior-art techniques.
These and other objects are achieved in a method as set forth in the opening paragraph above, which method is characterized by the following steps:
The method according to the current invention has a number of striking advantages. For example:
In a particular embodiment of the current invention:
It should be explicitly noted in the embodiment just discussed, but also in other embodiments of the invention (such as the next two to be discussed below), that the vector employed in the current invention can be multiplied by one or more proportionality constants without affecting the crux of the invention. For example, the position of a (particle) radiation spot on a segmented detector (relative to an elected origin) is of itself a vector. However, with due regard to the operating principle of the detector, such a vector can be assigned a specific physical meaning, e.g. an electrostatic potential field gradient (electric field), which is also a vector. Mathematically, it makes no difference which vector is integrated two-dimensionally: one can convert from one vector to the other (and from one integration result to the other) via a simple (resultant) proportionality constant.
In an alternative embodiment of the current invention which may be regarded as a refinement (or “higher-resolution” version) of the previously discussed embodiment the following applies:
In yet another embodiment of the present invention, the employed detector is a Position-Sensitive Detector (PSD). Such detectors are available in different forms, such as:
As regards the two-dimensional integration operation prescribed by the current invention, the area of Vector Field Integration (a sub-field of Vector Calculus) provides the basic mathematical framework needed to appropriately process the vector field distilled from the employed segmented detector. In specific instances where the vector concerned is a gradient (which allows certain simplifications to be assumed vis-à-vis its integrability), then the techniques of the more specific area of Gradient Field Integration can be applied. Examples of algorithms that can be used to perform such Gradient Field Integration on acquired image data can, for example, be gleaned from the following articles in technical journals, which concern themselves with (photon-optical) Machine Vision/Photometric Stereo issues:
Once the two-dimensional vector field integration according to the current invention has been conducted, the resulting “raw” integrated vector field image can, if so desired, be post-processed (i.e. “polished up”) by subjecting it to further mathematical manipulation. Such manipulation can, for example, involve at least one operation selected from the group comprising:
where:
where:
In the Fourier domain, this becomes:
The skilled artisan will understand that, in general, techniques (i)-(iii) described here may be applied individually or in combination.
In another embodiment of the present invention, the inventive integrated vector field image referred to above is further manipulated by subjecting it to a Laplacian operation. The Laplacian operator is a second-order differential operator that takes the following form in two-dimensional Euclidean space, when performed on a function ƒ:
In the current case, the entity ƒ may be either the “raw” integrated vector field image (iVF) produced by the invention, or a post-processed integrated vector field image (PiVF) as referred to in the previous embodiment. Performing this Laplacian operation produces an image (LiVF or LPiVF) that, for some features, can yield improved definition (though at the possible expense of other image qualities). See the examples given below in Embodiments 4 and 5, for instance. It should be noted that performing a Laplacian operation as described here may, if desired/necessary, be accompanied by multiplication by a proportionality/scaling/correction constant, such as −1 for example; this can, for instance, be done so as to make the result of the Laplacian operation more consistent with a certain physical interpretation.
It should be explicitly noted that the current invention is radically different to the so-called Differential Phase Contrast (DPC) technique. In the DPC method, a four-quadrant detector is used to produce scalar difference images such as the S1-S3 or S2-S4 images shown in
The invention will now be elucidated in more detail on the basis of exemplary embodiments and the accompanying schematic drawings, in which:
In the Figures, where pertinent, corresponding parts may be indicated using corresponding reference symbols. It should be noted that, in general, the Figures are not to scale.
The sample S is held on a sample holder 10 than can be positioned in multiple degrees of freedom by a positioning device (stage) 12; for example, the sample holder 10 may comprise a finger that can be moved (inter alia) in the XY plane (see the depicted Cartesian coordinate system). Such movement allows different regions of the sample S to be irradiated/imaged/inspected by the electron beam traveling along axis 8 (in the −Z direction) (and/or allows scanning motion to be performed, as an alternative to beam scanning). An optional cooling device 14 is in intimate thermal contact with the supporting device 10, and is capable of maintaining the latter at cryogenic temperatures, e.g. using a circulating cryogenic coolant to achieve and maintain a desired low temperature.
The focused electron beam traveling along axis 8 will interact with the sample S in such a manner as to cause various types of “stimulated” radiation to be emitted from the sample S, including (for example) secondary electrons, backscattered electrons, X-rays and optical radiation (cathodoluminescence); if desired, one or more of these radiation types can be detected with the aid of detector 22, which might be a combined scintillator/photomultiplier or EDX (Energy-Dispersive X-Ray Spectroscopy) detector, for instance. However, of predominant interest in the current invention are electrons that traverse (pass through) the sample S, emerge from it and continue to propagate (substantially, though generally with some deflection/scattering) along axis 8. Such transmitted electrons enter an imaging system (combined objective/projection lens) 24, which will generally comprise a variety of electrostatic/magnetic lenses, deflectors, correctors (such as stigmators), etc. In normal (non-scanning) TEM mode, this imaging system 24 can focus the transmitted electrons onto a fluorescent screen 26, which, if desired, can be retracted/withdrawn (as schematically indicated by arrows 28) so as to get it out of the way of axis 8. An image of (part of) the sample S will be formed by imaging system 24 on screen 26, and this may be viewed through viewing port 30 located in a suitable portion of the wall 2. The retraction mechanism for screen 26 may, for example, be mechanical and/or electrical in nature, and is not depicted here.
As an alternative to viewing an image on screen 26, one can instead make use of electron detector D, particularly in STEM mode. To this end, adjuster lens 24′ can be enacted so as to shift the focus of the electrons emerging from imaging system 24 and re-direct/focus them onto detector D (rather than the plane of retracted screen 26: see above). At detector D, the electrons can form an image (or diffractogram) that can be processed by controller 50 and displayed on a display device (not depicted), such as a flat panel display, for example. In STEM mode, an output from detector D can be recorded as a function of (X,Y) scanning beam position on the sample S, and an image can be constructed that is a “map” of detector output as a function of X,Y. The skilled artisan will be very familiar with these various possibilities, which require no further elucidation here.
Note that the controller (computer processor) 50 is connected to various illustrated components via control lines (buses) 50′. This controller 50 can provide a variety of functions, such as synchronizing actions, providing setpoints, processing signals, performing calculations, and displaying messages/information on a display device (not depicted). Needless to say, the (schematically depicted) controller 50 may be (partially) inside or outside the enclosure 2, and may have a unitary or composite structure, as desired. The skilled artisan will understand that the interior of the enclosure 2 does not have to be kept at a strict vacuum; for example, in a so-called “Environmental STEM”, a background atmosphere of a given gas is deliberately introduced/maintained within the enclosure 2.
In the context of the current invention, the following additional points deserve further elucidation:
As set forth above, a measured vector field {tilde over (E)}(x,y)=({tilde over (E)}x (x,y), {tilde over (E)}y(x,y))T can (for example) be derived at each coordinate point (x,y) from detector segment differences using the expressions:
where, for simplicity, spatial indexing (x,y) in the scalar fields {tilde over (E)}x, {tilde over (E)}y and Si=1, . . . , 4, has been omitted, and where superscript T denotes the transpose of a matrix.
It is known from the theory of STEM contrast formation that {tilde over (E)} is a measurement of the actual electric field E in an area of interest of the imaged specimen. This measurement is inevitably corrupted by noise and distortions caused by imperfections in optics, detectors, electronics, etc. From basic electromagnetism, it is known that the electrostatic potential function φ(x,y) [also referred to below as the potential map] is related to the electric field by:
E=−∇φ (3)
The goal here is to obtain the potential map at each scanned location of the specimen. But the measured electric field in its noisy form {tilde over (E)} will most likely not be “integrable”, i.e. cannot be derived from a smooth potential function by the gradient operator. The search for an estimate {circumflex over (φ)} of the potential map given the noisy measurements {tilde over (E)} can be formulated as a fitting problem, resulting in the functional minimization of objective function J defined as:
J(φ)=∫∫∥(−∇φ)−{tilde over (E)}∥2dxdy=∫∫∥∇φ+{tilde over (E)}∥2dxdy (4)
where
One is essentially looking for the closest fit to the measurements, in the least squares sense, of gradient fields derived from smooth potential functions φ.
To be at the sought minimum of J one must satisfy the Euler-Lagrange equation:
which can be expanded to:
finally resulting in:
which is the Poisson equation that one needs to solve to obtain Cp.
Poisson Solvers
Using finite differences for the derivatives in (7) one obtains:
where Δ is the so-called grid step size (assumed here to be equal in the x and y directions). The right side quantity in (8) is known from measurements and will be lumped together in a term ρi,j to simplify notation:
which, after rearranging, results in:
φi−1,j+φi,j−1−4φi,j+φi,j+1+φi+1,j=Δ2ρi,j (10)
for i=2, . . . , N−1 and j=2, . . . , M−1, with (N,M) the dimensions of the image to be reconstructed.
The system in (10) leads to the matrix formulation:
Lφ=ρ (11)
where φ and ρ represent the vector form of the potential map and measurements, respectively (the size of these vectors is N×M, which is the size of the image). The so-called Laplacian matrix L is of dimensions (N×M)2 but is highly sparse and has a special form called “tridiagonal with fringes” for the discretization scheme used above. So-called Dirichlet and Neumann boundary conditions are commonly used to fix the values of {circumflex over (φ)} at the edges of the potential map.
The linear system of (11) tends to be very large for typical STEM images, and will generally be solved using numerical methods, such as the bi-conjugate gradient method. Similar approaches have previously been used in topography reconstruction problems, as discussed, for example, in the journal article by Ruggero Pintus, Simona Podda and Massimo Vanzi, 14th European Microscopy Congress, Aachen, Germany, pp. 597-598, Springer (2008). One should note that other forms of discretization of the derivatives can be used in the previously described approach, and that the overall technique is conventionally known as the Poisson solver method. A specific example of such a method is the so-called multi-grid Poisson solver, which is optimized to numerically solve the Poisson equation starting from a coarse mesh/grid and proceeding to a finer mesh/grid, thus increasing integration speed.
Basis Function Reconstruction
Another approach to solving (7) is to use the so-called Frankot-Chellapa algorithm presented in the above-mentioned journal article by Frankot and Chellappa, which was previously employed for depth reconstruction from photometric stereo images. Adapting this method to the current problem, one can reconstruct the potential map by projecting the derivatives into the space-integrable Fourier basis functions. In practice, this is done by applying the Fourier Transform FT(•) to both sides of (7) to obtain:
(ωx2+ωy2)FT(φ)=−√{square root over (−1)}(ωxFT({tilde over (E)}x)+ωyFT({tilde over (E)}y)) (12)
from which {circumflex over (φ)} can be obtained by Inverse Fourier Transform (IFT):
The forward and inverse transforms can be implemented using the so-called Discrete Fourier Transform (DFT), in which case the assumed boundary conditions are periodic. Alternatively, one can use the so-called Discrete Sine Transform (DST), which corresponds to the use of the Dirichlet boundary condition (φ=0 at the boundary). One can also use the so-called Discrete Cosine Transform (DCT), corresponding to the use of the Neumann boundary conditions (∇φ·n=0 at the boundary, n being the normal vector at the given boundary location).
Generalizations and Improved Solutions
While working generally well, the Poisson solver and Basis Function techniques can be enhanced further by methods that take into account sharp discontinuities in the data (outliers). For that purpose, the objective function J can be modified to incorporate a different residual error R (in (4), the residual error was R(v)=∥v∥2). One can for example use exponents of less than two including so-called Lp norm-based objective functions:
J(φ)=∫∫R(−∇φ,{tilde over (E)})dxdy=∫∫∥(−∇φ)−{tilde over (E)}∥1/pdxdy,p≧1 (14)
The residual can also be chosen from the set of functions typically used in so-called M-estimators (a commonly used class of robust estimators). In this case, R can be chosen from among functions such as so-called Huber, Cauchy, and Tuckey functions. Again, the desired result from this modification of the objective function will be to avoid overly smooth reconstructions and to account more accurately for real/physical discontinuities in the datasets. Another way of achieving this is to use anisotropic weighting functions wx and wy in J:
J(φ)=∫∫wx(εxk-1)(−φx−{tilde over (E)}x)2+wy(εyk-1)(−φy−{tilde over (E)}y)2dxdy (15)
where the weight functions depend on the residuals:
R(εxk-1)=R(−φxk-1,{tilde over (E)}x) and R(εyk-1)=R(−φyk-1,{tilde over (E)}y) (15a)
at iteration k−1.
In the above-mentioned journal article by Agrawal, Chellappa and Raskar, it was shown that, for the problem of depth reconstruction from photometric stereo images, the use of such anisotropic weights, which can be either binary or continuous, leads to improved results in the depth map recovery process.
In another approach, one can also apply a diffusion tensor D to the vector fields ∇φ and {tilde over (E)} with the aim of smoothing the data while preserving discontinuities during the process of solving for {circumflex over (φ)}, resulting in the modification of (4) into:
J(φ)=∫∫ƒD(−∇φ)−D({tilde over (E)})∥2dxdy (16)
Finally, regularization techniques can be used to restrict the solution space. This is generally done by adding penalty functions in the formulation of the objective criterion J such as follows:
J(φ)=∫∫[∥(−∇φ)−{tilde over (E)}∥2+λƒ(∇φ)]dxdy (17)
The regularization function ƒ(∇φ) can be used to impose a variety of constraints on φ for the purpose of stabilizing the convergence of the iterative solution. It can also be used to incorporate into the optimization process prior knowledge about the sought potential field or other specimen/imaging conditions.
In the left-hand sub-Figure, different types of detector D are schematically illustrated. In particular:
In contrast, in the right-hand sub-Figure, essentially the entire angular spread of OF is captured and used by the detector D (which is here depicted as being a four-quadrant detector, with two quadrants Q1 and Q3 illustrated). The dashed line indicates a non-deflected/non-scattered “reference” cone of flux OF′ (essentially corresponding to that causing beam footprint B′ in
Normally, capturing such a large angular range of the output flux OF [situation (B)] would result in a contrast-less image; hence the piecemeal approach used in the prior art [situation (A)], which is, however, highly wasteful, in that it discards large portions of OF at any given time. The current invention nevertheless allows the highly efficient flux collection scenario in situation (B) to be used to produce a contrast-rich image, thanks to the innovative vector field integration procedure prescribed by the present inventors.
Left: Conventional (non-scanning) TEM (CTEM) images, depicting (from top to bottom) an image series with respective defocus values of 0.8 μm, 1.2 μm, 2.0 μm, 2.4 μm and 4.9 μm. Note that the largest employed defocus gives the best contrast in this series. The field of view in each member of this series is ca. 45 nm×45 nm.
Right: An integrated vector field (iVF) image according to the current invention, which renders much more detail than any of the CTEM images. The field of view in this case is ca. 30 nm×30 nm.
Note that the total electron dose was the same in both cases (10 e/Å2—electrons per square Ångstrom), which is an extremely low dose typically stipulated for imaging biological samples under cryogenic conditions. Apart from the current invention, no STEM-based technique shows any meaningful signal under such low-dosage conditions.
Using a Position Sensitive Detector (PSD) and measuring a thin, non-magnetic sample, one obtains (by definition) the vector field image components as components of the center of mass (COM) of the electron intensity distribution ID ({right arrow over (k)},{right arrow over (r)}p) at the detector plane:
IxCOM({right arrow over (r)}p)=∫∫−∞∞kxID({right arrow over (k)},{right arrow over (r)}p)d2{right arrow over (k)} IyCOM({right arrow over (r)}p)=∫∫−∞∞kyID({right arrow over (k)},{right arrow over (r)}p)d2{right arrow over (k)} (18)
where {right arrow over (r)}p represents position of the probe (focused electron beam) impinging upon the sample, and {right arrow over (k)}=(kx,ky) are coordinates in the detector plane. The full vector field image can then be formed as:
{right arrow over (ICOM)}({right arrow over (r)}p)=IxCOM({right arrow over (r)}p)·{right arrow over (x)}0+IyCOM({right arrow over (r)}p)·{right arrow over (y)}0 (19)
where {right arrow over (x)}0 and {right arrow over (y)}0 are unit vectors in two perpendicular directions.
The electron intensity distribution at the detector is given by:
ID({right arrow over (k)},{right arrow over (r)}p)=|{ψin({right arrow over (r)}−{right arrow over (r)}p)eiφ({right arrow over (r)})}({right arrow over (k)})|2 (20)
where ψin({right arrow over (r)}−{right arrow over (r)}p) is the impinging electron wave (i.e. the probe) illuminating the sample at position {right arrow over (r)}p and eiφ({right arrow over (r)}) is the transmission function of the sample. The phase φ({right arrow over (r)}) is proportional to the sample's inner electrostatic potential field. Imaging φ({right arrow over (r)}) is the ultimate goal of any electron microscopy imaging technique. Expression (19) can be re-written as:
where {right arrow over (E)}({right arrow over (r)})=−∇(φ({right arrow over (r)}) is the inner electric field of the sample—which is the negative gradient of the electrostatic potential field of the sample—and the operator “*” denotes cross-correlation. It is evident that the obtained vector field image {right arrow over (ICOM)}({right arrow over (r)}p) directly represents the inner electric field {right arrow over (E)}({right arrow over (r)}) of the sample. Its components are set forth in (18) above. Next, an integration step in accordance with the current invention is performed, as follows:
using any arbitrary path l. This arbitrary path is allowed because, in the case of non-magnetic samples, the only field is the electric field, which is a conservative vector field. Numerically this can be performed in many ways (see above). Analytically it can be worked out by introducing (21) into (22), yielding:
It is clear that, with this proposed integration step, one obtains a scalar field image that directly represents φ({right arrow over (r)}), which is the preferred object in electron microscopy.
CPM imagery is often rather noisy, e.g. due to dose limitations. Subtracting two noisy signals so as to obtain the “subtractive” or “gradient” images referred to above (see
∫∫|∇φ+{right arrow over (E)}|2dxdy
In this step, noise is regularized, as can be seen from the fact that the number of degrees of freedom (i.e. the number of pixels) is reduced by a factor of two (two images are combined into one resultant image). When the resulting potential is then differentiated, the number of degrees of freedom stays the same, and hence the resulting diVF (or dPiVF) images have less noise than the original images. The part of the noise that was removed is the part that cannot be written as the gradient of a scalar potential.
It should be noted that the single differentiation (∇; nabla/del) alluded to here produces a vector result—with, for example, X and Y components in the case of a Cartesian coordinate system. These components can, if desired, be calculated separately from one another, using partial differentiation
An example of this technique is shown in
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14156356 | Feb 2014 | EP | regional |
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20070194225 | Zorn | Aug 2007 | A1 |
20130193322 | Blackburn | Aug 2013 | A1 |
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Entry |
---|
Scheinfein, M. R., et al., “Scanning electron microscopy with polarization analysis (SEMPA)”, Review of Scientific Instruments, 1990, pp. 2501-2526, vol. 61, No. 10, Melville, NY, US. |
Nomizu, S., et al., “Reconstruction of Magnetic Recording Spatial Pattern on the Tape Media by Reflection Electron Beam Tomography”, IEEE Transactions on Magnetics, 1997, pp. 4032-4034, vol. 33, No. 5, New York, NY, US. |
Yajima, Y., et al., “Scanning Lorentz electron microscope with high resolution and observation of bit profiles recorded on sputtered longitudinal media (invited)”, Journal of Applied Physics, 1993, pp. 5811-5815, vol. 73, No. 10, US. |
Escovitz, W. H., et al., “Scanning Transmission Ion Microscope with a Field Ion Source,” Proc. Nat. Acad. Sci., vol. 72, No. 5, pp. 1826-1828, 1975 USA. |
Unknown, http://en.wikipedia.org/wiki/Electron—microscope, accessed May 22, 2015. |
Unknown, http://en.wikipedia.org/wiki/Scanning—electron—microscope, accessed May 22, 2015. |
Unknown, http://en.wikipedia.org/wiki/Transmission—electron—microscopy, accessed May 22, 2015. |
Unknown, http://en.wikipedia.org/wiki/Scanning—Helium—Ion—Microscope, accessed May 22, 2015. |
Unknown, http://en.wikipedia.org/wiki/Electron—tomography, accessed May 22, 2015. |
Unknown, http://en.wikipedia.org/wiki/HAADF, accessed May 22, 2015. |
Unknown, http://en.wikipedia.org/wiki/Position—sensitive—device, accessed May 22, 2015. |
Unknown, http://en.wikipedia.org/wiki/Vector—calculus, accessed May 22, 2015. |
Pintus, R., et al., “3D Sculptures From SEM Images,” EMC 2008 European Microscopy Congress Sep. 1-5, 2008, Aachen, Germany, pp. 597-598. |
Unknown, http://en.wikipedia.org/wiki/Scanning—transmission—electron—microscopy, accessed May 22, 2015. |
Frankot, Robert T., et al., “A Method for Enforcing Integrability in Shape from Shading Algorithms,” IEEE Transacations on Pattern Analysis and Machine Intelligence, vol. 10, No. 4, pp. 439-451, 1988. |
Aggarwal, Guarav, et al., “Face Recognition in the Presence of Multiple Illumination Sources,” Computer Vision, Tenth IEEE International Conference, vol. 2, pp. 1169-1176, 2005, Bejing, China. |
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20150243474 A1 | Aug 2015 | US |